NCERT Solutions for Class 12 Maths Chapter 5 – Continuity and Differentiability Ex 5.1
Page No 159:
Question 1:
Prove that the function
is continuous at![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6748/Chapter%205_html_48cc118a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6748/Chapter%205_html_1e328cf0.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6748/Chapter%205_html_48cc118a.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6748/Chapter%205_html_a69856a.gif)
Therefore, f is continuous at x = 0
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6748/Chapter%205_html_m3baa23bc.gif)
Therefore, f is continuous at x = −3
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6748/Chapter%205_html_75094717.gif)
Therefore, f is continuous at x = 5
Question 2:
Examine the continuity of the function
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6751/Chapter%205_html_m13c6d82c.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6751/Chapter%205_html_266c3190.gif)
Thus, f is continuous at x = 3
Question 3:
Examine the following functions for continuity.
(a)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_m63a64d7c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_m676cbc31.gif)
(c)
(d) ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_m2bcad135.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_m7502a14e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_m2bcad135.gif)
Answer:
(a) The given function is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_m63a64d7c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_m63a64d7c.gif)
It is evident that f is defined at every real number k and its value at k is k − 5.
It is also observed that, ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_m4a84a954.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_m4a84a954.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_13958887.gif)
Hence, f is continuous at every real number and therefore, it is a continuous function.
(b) The given function is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_m676cbc31.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_m676cbc31.gif)
For any real number k ≠ 5, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_50a3ea5c.gif)
Hence, f is continuous at every point in the domain of f and therefore, it is a continuous function.
(c) The given function is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_m7502a14e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_m7502a14e.gif)
For any real number c ≠ −5, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_m3f3d37a6.gif)
Hence, f is continuous at every point in the domain of f and therefore, it is a continuous function.
(d) The given function is ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_2c77b583.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_2c77b583.gif)
This function f is defined at all points of the real line.
Let c be a point on a real line. Then, c < 5 or c = 5 or c > 5
Case I: c < 5
Then, f (c) = 5 − c
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_m1296b13d.gif)
Therefore, f is continuous at all real numbers less than 5.
Case II : c = 5
Then, ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_m37a44f8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_m37a44f8.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_c8fc2c0.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_15e897ad.gif)
Therefore, f is continuous at x = 5
Case III: c > 5
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_m45cd474e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6754/Chapter%205_html_m3078a2af.gif)
Therefore, f is continuous at all real numbers greater than 5.
Hence, f is continuous at every real number and therefore, it is a continuous function.
Question 4:
Prove that the function
is continuous at x = n, where n is a positive integer.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6755/Chapter%205_html_m7cd8b8a0.gif)
Answer:
The given function is f (x) = xn
It is evident that f is defined at all positive integers, n, and its value at n is nn.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6755/Chapter%205_html_75e17cf3.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6755/Chapter%205_html_4dc28689.gif)
Therefore, f is continuous at n, where n is a positive integer.
Question 5:
Is the function f defined by
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6756/Chapter%205_html_m43306763.gif)
continuous at x = 0? At x = 1? At x = 2?
Answer:
The given function f is ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6756/Chapter%205_html_m43306763.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6756/Chapter%205_html_m43306763.gif)
At x = 0,
It is evident that f is defined at 0 and its value at 0 is 0.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6756/Chapter%205_html_1a9d0cd6.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6756/Chapter%205_html_m5835d82a.gif)
Therefore, f is continuous at x = 0
At x = 1,
f is defined at 1 and its value at 1 is 1.
The left hand limit of f at x = 1 is,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6756/Chapter%205_html_58751911.gif)
The right hand limit of f at x = 1 is,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6756/Chapter%205_html_m12aff82a.gif)
Therefore, f is not continuous at x = 1
At x = 2,
f is defined at 2 and its value at 2 is 5.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6756/Chapter%205_html_2480fbf6.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6756/Chapter%205_html_189a3d61.gif)
Therefore, f is continuous at x = 2
Question 6:
Find all points of discontinuity of f, where f is defined by
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6758/Chapter%205_html_6f5d7057.gif)
Answer:
The given function f is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6758/Chapter%205_html_6f5d7057.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6758/Chapter%205_html_6f5d7057.gif)
It is evident that the given function f is defined at all the points of the real line.
Let c be a point on the real line. Then, three cases arise.
(i) c < 2
(ii) c > 2
(iii) c = 2
Case (i) c < 2
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6758/Chapter%205_html_m78776ee0.gif)
Therefore, f is continuous at all points x, such that x < 2
Case (ii) c > 2
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6758/Chapter%205_html_m5ef9a613.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6758/Chapter%205_html_m3078a2af.gif)
Therefore, f is continuous at all points x, such that x > 2
Case (iii) c = 2
Then, the left hand limit of f at x = 2 is,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6758/Chapter%205_html_mbe21158.gif)
The right hand limit of f at x = 2 is,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6758/Chapter%205_html_6855250.gif)
It is observed that the left and right hand limit of f at x = 2 do not coincide.
Therefore, f is not continuous at x = 2
Hence, x = 2 is the only point of discontinuity of f.
Question 7:
Find all points of discontinuity of f, where f is defined by
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6759/Chapter%205_html_m763140dc.gif)
Answer:
The given function f is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6759/Chapter%205_html_m66ddd76d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6759/Chapter%205_html_m66ddd76d.gif)
The given function f is defined at all the points of the real line.
Let c be a point on the real line.
Case I:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6759/Chapter%205_html_67637c17.gif)
Therefore, f is continuous at all points x, such that x < −3
Case II:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6759/Chapter%205_html_6d71b1df.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6759/Chapter%205_html_727680e3.gif)
Therefore, f is continuous at x = −3
Case III:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6759/Chapter%205_html_m2903fe5e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6759/Chapter%205_html_m3078a2af.gif)
Therefore, f is continuous in (−3, 3).
Case IV:
If c = 3, then the left hand limit of f at x = 3 is,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6759/Chapter%205_html_5941cb89.gif)
The right hand limit of f at x = 3 is,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6759/Chapter%205_html_29f0b36.gif)
It is observed that the left and right hand limit of f at x = 3 do not coincide.
Therefore, f is not continuous at x = 3
Case V:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6759/Chapter%205_html_79947981.gif)
Therefore, f is continuous at all points x, such that x > 3
Hence, x = 3 is the only point of discontinuity of f.
Question 8:
Find all points of discontinuity of f, where f is defined by
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6760/Chapter%205_html_m4ccee9e7.gif)
Answer:
The given function f is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6760/Chapter%205_html_m4ccee9e7.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6760/Chapter%205_html_m4ccee9e7.gif)
It is known that,![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6760/Chapter%205_html_5509af4.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6760/Chapter%205_html_5509af4.gif)
Therefore, the given function can be rewritten as
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6760/Chapter%205_html_m292cde6.gif)
The given function f is defined at all the points of the real line.
Let c be a point on the real line.
Case I:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6760/Chapter%205_html_m28f79be1.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6760/Chapter%205_html_m3078a2af.gif)
Therefore, f is continuous at all points x < 0
Case II:
If c = 0, then the left hand limit of f at x = 0 is,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6760/Chapter%205_html_3fa92674.gif)
The right hand limit of f at x = 0 is,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6760/Chapter%205_html_m4c329f7d.gif)
It is observed that the left and right hand limit of f at x = 0 do not coincide.
Therefore, f is not continuous at x = 0
Case III:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6760/Chapter%205_html_m4bef27dd.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6760/Chapter%205_html_m3078a2af.gif)
Therefore, f is continuous at all points x, such that x > 0
Hence, x = 0 is the only point of discontinuity of f.
Question 9:
Find all points of discontinuity of f, where f is defined by
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6761/Chapter%205_html_m3ec67b4a.gif)
Answer:
The given function f is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6761/Chapter%205_html_m3ec67b4a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6761/Chapter%205_html_m3ec67b4a.gif)
It is known that,![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6761/Chapter%205_html_m27c7ceea.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6761/Chapter%205_html_m27c7ceea.gif)
Therefore, the given function can be rewritten as
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6761/Chapter%205_html_2ac3ed3e.gif)
Let c be any real number. Then, ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6761/Chapter%205_html_3ef9fc07.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6761/Chapter%205_html_3ef9fc07.gif)
Also,![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6761/Chapter%205_html_378e5a68.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6761/Chapter%205_html_378e5a68.gif)
Therefore, the given function is a continuous function.
Hence, the given function has no point of discontinuity.
Question 10:
Find all points of discontinuity of f, where f is defined by
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6762/Chapter%205_html_m5cd13fd3.gif)
Answer:
The given function f is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6762/Chapter%205_html_m5cd13fd3.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6762/Chapter%205_html_m5cd13fd3.gif)
The given function f is defined at all the points of the real line.
Let c be a point on the real line.
Case I:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6762/Chapter%205_html_3501b6a4.gif)
Therefore, f is continuous at all points x, such that x < 1
Case II:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6762/Chapter%205_html_m4ba2db85.gif)
The left hand limit of f at x = 1 is,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6762/Chapter%205_html_4d9c2e78.gif)
The right hand limit of f at x = 1 is,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6762/Chapter%205_html_m2b97949e.gif)
Therefore, f is continuous at x = 1
Case III:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6762/Chapter%205_html_m68ca6455.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6762/Chapter%205_html_m3078a2af.gif)
Therefore, f is continuous at all points x, such that x > 1
Hence, the given function f has no point of discontinuity.
Question 11:
Find all points of discontinuity of f, where f is defined by
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6763/Chapter%205_html_224340ea.gif)
Answer:
The given function f is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6763/Chapter%205_html_224340ea.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6763/Chapter%205_html_224340ea.gif)
The given function f is defined at all the points of the real line.
Let c be a point on the real line.
Case I:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6763/Chapter%205_html_m35d1b7b9.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6763/Chapter%205_html_m3078a2af.gif)
Therefore, f is continuous at all points x, such that x < 2
Case II:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6763/Chapter%205_html_m442e0a43.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6763/Chapter%205_html_6003b281.gif)
Therefore, f is continuous at x = 2
Case III:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6763/Chapter%205_html_m2052425d.gif)
Therefore, f is continuous at all points x, such that x > 2
Thus, the given function f is continuous at every point on the real line.
Hence, f has no point of discontinuity.
Question 12:
Find all points of discontinuity of f, where f is defined by
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6765/Chapter%205_html_21ccc991.gif)
Answer:
The given function f is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6765/Chapter%205_html_21ccc991.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6765/Chapter%205_html_21ccc991.gif)
The given function f is defined at all the points of the real line.
Let c be a point on the real line.
Case I:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6765/Chapter%205_html_2b68e9d.gif)
Therefore, f is continuous at all points x, such that x < 1
Case II:
If c = 1, then the left hand limit of f at x = 1 is,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6765/Chapter%205_html_m4a164abb.gif)
The right hand limit of f at x = 1 is,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6765/Chapter%205_html_6e433823.gif)
It is observed that the left and right hand limit of f at x = 1 do not coincide.
Therefore, f is not continuous at x = 1
Case III:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6765/Chapter%205_html_6d5b96ba.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6765/Chapter%205_html_m3078a2af.gif)
Therefore, f is continuous at all points x, such that x > 1
Thus, from the above observation, it can be concluded that x = 1 is the only point of discontinuity of f.
Question 13:
Is the function defined by
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6767/Chapter%205_html_m1d76af7c.gif)
a continuous function?
Answer:
The given function is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6767/Chapter%205_html_m1d76af7c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6767/Chapter%205_html_m1d76af7c.gif)
The given function f is defined at all the points of the real line.
Let c be a point on the real line.
Case I:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6767/Chapter%205_html_71e1a3a3.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6767/Chapter%205_html_m3078a2af.gif)
Therefore, f is continuous at all points x, such that x < 1
Case II:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6767/Chapter%205_html_m49f14e3.gif)
The left hand limit of f at x = 1 is,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6767/Chapter%205_html_m33dba451.gif)
The right hand limit of f at x = 1 is,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6767/Chapter%205_html_47b80a2.gif)
It is observed that the left and right hand limit of f at x = 1 do not coincide.
Therefore, f is not continuous at x = 1
Case III:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6767/Chapter%205_html_3afae62a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6767/Chapter%205_html_m3078a2af.gif)
Therefore, f is continuous at all points x, such that x > 1
Thus, from the above observation, it can be concluded that x = 1 is the only point of discontinuity of f.
Page No 160:
Question 14:
Discuss the continuity of the function f, where f is defined by
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6768/Chapter%205_html_20fb2c5f.gif)
Answer:
The given function is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6768/Chapter%205_html_20fb2c5f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6768/Chapter%205_html_20fb2c5f.gif)
The given function is defined at all points of the interval [0, 10].
Let c be a point in the interval [0, 10].
Case I:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6768/Chapter%205_html_m682944c5.gif)
Therefore, f is continuous in the interval [0, 1).
Case II:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6768/Chapter%205_html_m6367ab7.gif)
The left hand limit of f at x = 1 is,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6768/Chapter%205_html_m7a7b2037.gif)
The right hand limit of f at x = 1 is,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6768/Chapter%205_html_49b439a6.gif)
It is observed that the left and right hand limits of f at x = 1 do not coincide.
Therefore, f is not continuous at x = 1
Case III:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6768/Chapter%205_html_m6269866a.gif)
Therefore, f is continuous at all points of the interval (1, 3).
Case IV:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6768/Chapter%205_html_me0f6680.gif)
The left hand limit of f at x = 3 is,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6768/Chapter%205_html_m7ac0d075.gif)
The right hand limit of f at x = 3 is,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6768/Chapter%205_html_m590ecce0.gif)
It is observed that the left and right hand limits of f at x = 3 do not coincide.
Therefore, f is not continuous at x = 3
Case V:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6768/Chapter%205_html_m561a1438.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6768/Chapter%205_html_5a589017.gif)
Therefore, f is continuous at all points of the interval (3, 10].
Hence, f is not continuous at x = 1 and x = 3
Question 15:
Discuss the continuity of the function f, where f is defined by
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6769/Chapter%205_html_m24dbf833.gif)
Answer:
The given function is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6769/Chapter%205_html_m24dbf833.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6769/Chapter%205_html_m24dbf833.gif)
The given function is defined at all points of the real line.
Let c be a point on the real line.
Case I:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6769/Chapter%205_html_964708f.gif)
Therefore, f is continuous at all points x, such that x < 0
Case II:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6769/Chapter%205_html_4fe26b04.gif)
The left hand limit of f at x = 0 is,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6769/Chapter%205_html_15c64d62.gif)
The right hand limit of f at x = 0 is,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6769/Chapter%205_html_6c4c75cf.gif)
Therefore, f is continuous at x = 0
Case III:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6769/Chapter%205_html_m18bc350f.gif)
Therefore, f is continuous at all points of the interval (0, 1).
Case IV:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6769/Chapter%205_html_6b889dd5.gif)
The left hand limit of f at x = 1 is,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6769/Chapter%205_html_d825cb9.gif)
The right hand limit of f at x = 1 is,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6769/Chapter%205_html_30c158c5.gif)
It is observed that the left and right hand limits of f at x = 1 do not coincide.
Therefore, f is not continuous at x = 1
Case V:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6769/Chapter%205_html_m2bca5b0a.gif)
Therefore, f is continuous at all points x, such that x > 1
Hence, f is not continuous only at x = 1
Question 16:
Discuss the continuity of the function f, where f is defined by
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6773/Chapter%205_html_5a22c2c0.gif)
Answer:
The given function f is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6773/Chapter%205_html_5a22c2c0.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6773/Chapter%205_html_5a22c2c0.gif)
The given function is defined at all points of the real line.
Let c be a point on the real line.
Case I:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6773/Chapter%205_html_2d706456.gif)
Therefore, f is continuous at all points x, such that x < −1
Case II:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6773/Chapter%205_html_m5c4dc14e.gif)
The left hand limit of f at x = −1 is,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6773/Chapter%205_html_71dd50a6.gif)
The right hand limit of f at x = −1 is,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6773/Chapter%205_html_7e7fb68a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6773/Chapter%205_html_m2c8a5e24.gif)
Therefore, f is continuous at x = −1
Case III:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6773/Chapter%205_html_m1f17b7c9.gif)
Therefore, f is continuous at all points of the interval (−1, 1).
Case IV:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6773/Chapter%205_html_m441906ff.gif)
The left hand limit of f at x = 1 is,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6773/Chapter%205_html_m6ed05fd3.gif)
The right hand limit of f at x = 1 is,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6773/Chapter%205_html_518dec0f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6773/Chapter%205_html_2124cbf9.gif)
Therefore, f is continuous at x = 2
Case V:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6773/Chapter%205_html_m56801209.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6773/Chapter%205_html_5a589017.gif)
Therefore, f is continuous at all points x, such that x > 1
Thus, from the above observations, it can be concluded that f is continuous at all points of the real line.
Question 17:
Find the relationship between a and b so that the function f defined by
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6779/Chapter%205_html_7394646a.gif)
is continuous at x = 3.
Answer:
The given function f is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6779/Chapter%205_html_7394646a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6779/Chapter%205_html_7394646a.gif)
If f is continuous at x = 3, then
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6779/Chapter%205_html_m33fde032.gif)
Therefore, from (1), we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6779/Chapter%205_html_4ee69190.gif)
Therefore, the required relationship is given by,![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6779/Chapter%205_html_m11cd780a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6779/Chapter%205_html_m11cd780a.gif)
Question 18:
For what value of
is the function defined by
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6784/Chapter%205_html_m11cc021f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6784/Chapter%205_html_2ac5cb7a.gif)
continuous at x = 0? What about continuity at x = 1?
Answer:
The given function f is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6784/Chapter%205_html_2ac5cb7a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6784/Chapter%205_html_2ac5cb7a.gif)
If f is continuous at x = 0, then
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6784/Chapter%205_html_5b2103db.gif)
Therefore, there is no value of λ for which f is continuous at x = 0
At x = 1,
f (1) = 4x + 1 = 4 × 1 + 1 = 5
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6784/Chapter%205_html_m7a9447a8.gif)
Therefore, for any values of λ, f is continuous at x = 1
Question 19:
Show that the function defined by
is discontinuous at all integral point. Here
denotes the greatest integer less than or equal to x.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6788/Chapter%205_html_m40f6c85a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6788/Chapter%205_html_m2694cf9.gif)
Answer:
The given function is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6788/Chapter%205_html_m40f6c85a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6788/Chapter%205_html_m40f6c85a.gif)
It is evident that g is defined at all integral points.
Let n be an integer.
Then,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6788/Chapter%205_html_m207aa0cb.gif)
The left hand limit of f at x = n is,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6788/Chapter%205_html_m1c2be5f0.gif)
The right hand limit of f at x = n is,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6788/Chapter%205_html_m172b5fc1.gif)
It is observed that the left and right hand limits of f at x = n do not coincide.
Therefore, f is not continuous at x = n
Hence, g is discontinuous at all integral points.
Question 20:
Is the function defined by
continuous at x =
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6792/Item%2020_html_m3add3190.gif)
π?
Answer:
The given function is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6792/Item%2020_html_m3add3190.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6792/Item%2020_html_m3add3190.gif)
It is evident that f is defined at x =
π.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6792/Item%2020_html_m7c663ad3.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6792/Item%2020_html_m588c68e3.gif)
Therefore, the given function f is continuous at x = π
Question 21:
Discuss the continuity of the following functions.
(a) f (x) = sin x + cos x
(b) f (x) = sin x − cos x
(c) f (x) = sin x × cos x
Answer:
It is known that if g and h are two continuous functions, then
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6796/Chapter%205_html_m60aee736.gif)
It has to proved first that g (x) = sin x and h (x) = cos x are continuous functions.
Let g (x) = sin x
It is evident that g (x) = sin x is defined for every real number.
Let c be a real number. Put x = c + h
If x → c, then h → 0
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6796/Chapter%205_html_m40481c7a.gif)
Therefore, g is a continuous function.
Let h (x) = cos x
It is evident that h (x) = cos x is defined for every real number.
Let c be a real number. Put x = c + h
If x → c, then h → 0
h (c) = cos c
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6796/Chapter%205_html_m60ec81a5.gif)
Therefore, h is a continuous function.
Therefore, it can be concluded that
(a) f (x) = g (x) + h (x) = sin x + cos x is a continuous function
(b) f (x) = g (x) − h (x) = sin x − cos x is a continuous function
(c) f (x) = g (x) × h (x) = sin x × cos x is a continuous function
Question 22:
Discuss the continuity of the cosine, cosecant, secant and cotangent functions,
Answer:
It is known that if g and h are two continuous functions, then
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6802/Item%2022_html_b9dd9b.gif)
It has to be proved first that g (x) = sin x and h (x) = cos x are continuous functions.
Let g (x) = sin x
It is evident that g (x) = sin x is defined for every real number.
Let c be a real number. Put x = c + h
If x
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6802/Item%2022_html_m48851eb.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6802/Item%2022_html_m48851eb.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6802/Item%2022_html_m40481c7a.gif)
Therefore, g is a continuous function.
Let h (x) = cos x
It is evident that h (x) = cos x is defined for every real number.
Let c be a real number. Put x = c + h
If x ® c, then h ® 0
h (c) = cos c
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6802/Item%2022_html_m60ec81a5.gif)
Therefore, h (x) = cos x is continuous function.
It can be concluded that,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6802/Item%2022_html_m6892dbfe.gif)
Therefore, cosecant is continuous except at x = np, n Î Z
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6802/Item%2022_html_mb65bf10.gif)
Therefore, secant is continuous except at ![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6802/Item%2022_html_m1aadeded.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6802/Item%2022_html_m1aadeded.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6802/Item%2022_html_4fef99da.gif)
Therefore, cotangent is continuous except at x = np, n Î Z
Question 23:
Find the points of discontinuity of f, where
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6807/Chapter%205_html_m324e9506.gif)
Answer:
The given function f is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6807/Chapter%205_html_m324e9506.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6807/Chapter%205_html_m324e9506.gif)
It is evident that f is defined at all points of the real line.
Let c be a real number.
Case I:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6807/Chapter%205_html_m7f62ef9c.gif)
Therefore, f is continuous at all points x, such that x < 0
Case II:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6807/Chapter%205_html_30eff037.gif)
Therefore, f is continuous at all points x, such that x > 0
Case III:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6807/Chapter%205_html_2b68db9d.gif)
The left hand limit of f at x = 0 is,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6807/Chapter%205_html_m7ab148d0.gif)
The right hand limit of f at x = 0 is,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6807/Chapter%205_html_mae478ee.gif)
Therefore, f is continuous at x = 0
From the above observations, it can be concluded that f is continuous at all points of the real line.
Thus, f has no point of discontinuity.
Question 24:
Determine if f defined by
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6812/Chapter%205_html_1bf52e85.gif)
is a continuous function?
Answer:
The given function f is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6812/Chapter%205_html_1bf52e85.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6812/Chapter%205_html_1bf52e85.gif)
It is evident that f is defined at all points of the real line.
Let c be a real number.
Case I:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6812/Chapter%205_html_5690438c.gif)
Therefore, f is continuous at all points x ≠ 0
Case II:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6812/Chapter%205_html_73e8f29.gif)
![](https://img-nm.mnimgs.com/img/study_content/content_ck_images/images/Selection_007(10).png)
⇒-x2≤x2sin1x≤x2
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6812/Chapter%205_html_m473a2a2d.gif)
![](https://img-nm.mnimgs.com/img/study_content/content_ck_images/images/Selection_009(12).png)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6812/Chapter%205_html_m473a2a2d.gif)
Therefore, f is continuous at x = 0
From the above observations, it can be concluded that f is continuous at every point of the real line.
Thus, f is a continuous function.
Page No 161:
Question 25:
Examine the continuity of f, where f is defined by
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6817/Chapter%205_html_75920e75.gif)
Answer:
The given function f is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6817/Chapter%205_html_75920e75.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6817/Chapter%205_html_75920e75.gif)
It is evident that f is defined at all points of the real line.
Let c be a real number.
Case I:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6817/Chapter%205_html_31cd584e.gif)
Therefore, f is continuous at all points x, such that x ≠ 0
Case II:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6817/Chapter%205_html_m14ad73a7.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6817/Chapter%205_html_c367a60.gif)
Therefore, f is continuous at x = 0
From the above observations, it can be concluded that f is continuous at every point of the real line.
Thus, f is a continuous function.
Question 26:
Find the values of k so that the function f is continuous at the indicated point.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6819/Item%2026_html_50475cb3.gif)
Answer:
The given function f is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6819/Item%2026_html_338e09a5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6819/Item%2026_html_338e09a5.gif)
The given function f is continuous at
, if f is defined at
and if the value of the f at
equals the limit of f at
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6819/Item%2026_html_m47fe309a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6819/Item%2026_html_m47fe309a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6819/Item%2026_html_m47fe309a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6819/Item%2026_html_m47fe309a.gif)
It is evident that f is defined at
and![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6819/Item%2026_html_m46f41e75.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6819/Item%2026_html_m47fe309a.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6819/Item%2026_html_m46f41e75.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6819/Item%2026_html_46e34999.gif)
Therefore, the required value of k is 6.
Question 27:
Find the values of k so that the function f is continuous at the indicated point.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6822/Chapter%205_html_m2d83caea.gif)
Answer:
The given function is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6822/Chapter%205_html_38fec36d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6822/Chapter%205_html_38fec36d.gif)
The given function f is continuous at x = 2, if f is defined at x = 2 and if the value of f at x = 2 equals the limit of f atx = 2
It is evident that f is defined at x = 2 and![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6822/Chapter%205_html_a656feb.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6822/Chapter%205_html_a656feb.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6822/Chapter%205_html_79a9c467.gif)
Therefore, the required value of
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6822/Chapter%205_html_m78d73418.gif)
Question 28:
Find the values of k so that the function f is continuous at the indicated point.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6826/Item%2028_html_66eb254b.gif)
Answer:
The given function is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6826/Item%2028_html_12f6ac2f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6826/Item%2028_html_12f6ac2f.gif)
The given function f is continuous at x = p, if f is defined at x = p and if the value of f at x = p equals the limit of f atx = p
It is evident that f is defined at x = p and![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6826/Item%2028_html_m501255cb.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6826/Item%2028_html_m501255cb.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6826/Item%2028_html_m1ce988b8.gif)
Therefore, the required value of![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6826/Item%2028_html_m563bdc98.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6826/Item%2028_html_m563bdc98.gif)
Question 29:
Find the values of k so that the function f is continuous at the indicated point.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6831/Chapter%205_html_1b0651b3.gif)
Answer:
The given function f is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6831/Chapter%205_html_m71365e67.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6831/Chapter%205_html_m71365e67.gif)
The given function f is continuous at x = 5, if f is defined at x = 5 and if the value of f at x = 5 equals the limit of f atx = 5
It is evident that f is defined at x = 5 and![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6831/Chapter%205_html_a7f4fd3.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6831/Chapter%205_html_a7f4fd3.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6831/Chapter%205_html_19cdbff6.gif)
Therefore, the required value of![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6831/Chapter%205_html_m5e28c8c3.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6831/Chapter%205_html_m5e28c8c3.gif)
Question 30:
Find the values of a and b such that the function defined by
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6833/Chapter%205_html_m49453141.gif)
is a continuous function.
Answer:
The given function f is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6833/Chapter%205_html_m49453141.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6833/Chapter%205_html_m49453141.gif)
It is evident that the given function f is defined at all points of the real line.
If f is a continuous function, then f is continuous at all real numbers.
In particular, f is continuous at x = 2 and x = 10
Since f is continuous at x = 2, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6833/Chapter%205_html_m6a67a506.gif)
Since f is continuous at x = 10, we obtain
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6833/Chapter%205_html_m62db6148.gif)
On subtracting equation (1) from equation (2), we obtain
8a = 16
⇒ a = 2
By putting a = 2 in equation (1), we obtain
2 × 2 + b = 5
⇒ 4 + b = 5
⇒ b = 1
Therefore, the values of a and b for which f is a continuous function are 2 and 1 respectively.
Question 31:
Show that the function defined by f (x) = cos (x2) is a continuous function.
Answer:
The given function is f (x) = cos (x2)
This function f is defined for every real number and f can be written as the composition of two functions as,
f = g o h, where g (x) = cos x and h (x) = x2
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6835/Chapter%205_html_2414acbb.gif)
It has to be first proved that g (x) = cos x and h (x) = x2 are continuous functions.
It is evident that g is defined for every real number.
Let c be a real number.
Then, g (c) = cos c
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6835/Chapter%205_html_m6d0b3abd.gif)
Therefore, g (x) = cos x is continuous function.
h (x) = x2
Clearly, h is defined for every real number.
Let k be a real number, then h (k) = k2
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6835/Chapter%205_html_m129f05e7.gif)
Therefore, h is a continuous function.
It is known that for real valued functions g and h,such that (g o h) is defined at c, if g is continuous at c and if f is continuous at g (c), then (f o g) is continuous at c.
Therefore,
is a continuous function.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6835/Chapter%205_html_71df85f5.gif)
Question 32:
Show that the function defined by
is a continuous function.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6839/Chapter%205_html_m67314a85.gif)
Answer:
The given function is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6839/Chapter%205_html_m67314a85.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6839/Chapter%205_html_m67314a85.gif)
This function f is defined for every real number and f can be written as the composition of two functions as,
f = g o h, where![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6839/Chapter%205_html_m252b46c6.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6839/Chapter%205_html_m252b46c6.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6839/Chapter%205_html_m400b359b.gif)
It has to be first proved that
are continuous functions.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6839/Chapter%205_html_m252b46c6.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6839/Chapter%205_html_5c7c3891.gif)
Clearly, g is defined for all real numbers.
Let c be a real number.
Case I:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6839/Chapter%205_html_m28e427f3.gif)
Therefore, g is continuous at all points x, such that x < 0
Case II:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6839/Chapter%205_html_7eaf7bf6.gif)
Therefore, g is continuous at all points x, such that x > 0
Case III:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6839/Chapter%205_html_m40b8e9ef.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6839/Chapter%205_html_7f384cbb.gif)
Therefore, g is continuous at x = 0
From the above three observations, it can be concluded that g is continuous at all points.
h (x) = cos x
It is evident that h (x) = cos x is defined for every real number.
Let c be a real number. Put x = c + h
If x → c, then h → 0
h (c) = cos c
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6839/Chapter%205_html_m60ec81a5.gif)
Therefore, h (x) = cos x is a continuous function.
It is known that for real valued functions g and h,such that (g o h) is defined at c, if g is continuous at c and if f is continuous at g (c), then (f o g) is continuous at c.
Therefore,
is a continuous function.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6839/Chapter%205_html_m3735b0cc.gif)
Question 33:
Examine that
is a continuous function.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6845/Chapter%205_html_266e30fc.gif)
Answer:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6845/Chapter%205_html_4c157e2.gif)
This function f is defined for every real number and f can be written as the composition of two functions as,
f = g o h, where![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6845/Chapter%205_html_m404b820b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6845/Chapter%205_html_m404b820b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6845/Chapter%205_html_4d6be309.gif)
It has to be proved first that
are continuous functions.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6845/Chapter%205_html_m404b820b.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6845/Chapter%205_html_5c7c3891.gif)
Clearly, g is defined for all real numbers.
Let c be a real number.
Case I:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6845/Chapter%205_html_m28e427f3.gif)
Therefore, g is continuous at all points x, such that x < 0
Case II:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6845/Chapter%205_html_7eaf7bf6.gif)
Therefore, g is continuous at all points x, such that x > 0
Case III:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6845/Chapter%205_html_m40b8e9ef.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6845/Chapter%205_html_7f384cbb.gif)
Therefore, g is continuous at x = 0
From the above three observations, it can be concluded that g is continuous at all points.
h (x) = sin x
It is evident that h (x) = sin x is defined for every real number.
Let c be a real number. Put x = c + k
If x → c, then k → 0
h (c) = sin c
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6845/Chapter%205_html_4d11afc.gif)
Therefore, h is a continuous function.
It is known that for real valued functions g and h,such that (g o h) is defined at c, if g is continuous at c and if f is continuous at g (c), then (f o g) is continuous at c.
Therefore,
is a continuous function.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6845/Chapter%205_html_m6ee2e49d.gif)
Question 34:
Find all the points of discontinuity of f defined by
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6849/Chapter%205_html_163ea7b7.gif)
Answer:
The given function is![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6849/Chapter%205_html_163ea7b7.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6849/Chapter%205_html_163ea7b7.gif)
The two functions, g and h, are defined as
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6849/Chapter%205_html_m5b88ece7.gif)
Then, f = g − h
The continuity of g and h is examined first.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6849/Chapter%205_html_5c7c3891.gif)
Clearly, g is defined for all real numbers.
Let c be a real number.
Case I:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6849/Chapter%205_html_m28e427f3.gif)
Therefore, g is continuous at all points x, such that x < 0
Case II:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6849/Chapter%205_html_7eaf7bf6.gif)
Therefore, g is continuous at all points x, such that x > 0
Case III:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6849/Chapter%205_html_m40b8e9ef.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6849/Chapter%205_html_7f384cbb.gif)
Therefore, g is continuous at x = 0
From the above three observations, it can be concluded that g is continuous at all points.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6849/Chapter%205_html_2697ef2d.gif)
Clearly, h is defined for every real number.
Let c be a real number.
Case I:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6849/Chapter%205_html_602d838d.gif)
Therefore, h is continuous at all points x, such that x < −1
Case II:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6849/Chapter%205_html_m7a5a8238.gif)
Therefore, h is continuous at all points x, such that x > −1
Case III:
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6849/Chapter%205_html_a45610c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6849/Chapter%205_html_m56c2d36f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/234/6849/Chapter%205_html_m2f351985.gif)
Therefore, h is continuous at x = −1
From the above three observations, it can be concluded that h is continuous at all points of the real line.
g and h are continuous functions. Therefore, f = g − h is also a continuous function.
Therefore, f has no point of discontinuity.