Free Learning Resources
PW Books
Notes (Class 10-12)
PW Study Materials
Notes (Class 6-9)
Ncert Solutions
Govt Exams
Class 6th to 12th Online Courses
Govt Job Exams Courses
UPSC Coaching
Defence Exam Coaching
Gate Exam Coaching
Other Exams
Know about Physics Wallah
Physics Wallah is an Indian edtech platform that provides accessible & comprehensive learning experiences to students from Class 6th to postgraduate level. We also provide extensive NCERT solutions, sample paper, NEET, JEE Mains, BITSAT previous year papers & more such resources to students. Physics Wallah also caters to over 3.5 million registered students and over 78 lakh+ Youtube subscribers with 4.8 rating on its app.
We Stand Out because
We provide students with intensive courses with India’s qualified & experienced faculties & mentors. PW strives to make the learning experience comprehensive and accessible for students of all sections of society. We believe in empowering every single student who couldn't dream of a good career in engineering and medical field earlier.
Our Key Focus Areas
Physics Wallah's main focus is to make the learning experience as economical as possible for all students. With our affordable courses like Lakshya, Udaan and Arjuna and many others, we have been able to provide a platform for lakhs of aspirants. From providing Chemistry, Maths, Physics formula to giving e-books of eminent authors like RD Sharma, RS Aggarwal and Lakhmir Singh, PW focuses on every single student's need for preparation.
What Makes Us Different
Physics Wallah strives to develop a comprehensive pedagogical structure for students, where they get a state-of-the-art learning experience with study material and resources. Apart from catering students preparing for JEE Mains and NEET, PW also provides study material for each state board like Uttar Pradesh, Bihar, and others
Copyright © 2025 Physicswallah Limited All rights reserved.
Thus, f is continuous at x = 3
(c)
Solution :
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.
Therefore, f is continuous at n, where n is a positive integer.
continuous at x = 0? At x = 1? At x = 2?
Solution :
The given function f is
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,
The right hand limit of f at x = 1 is,
Therefore, f is not continuous at x = 1
At x = 2,
f is defined at 2 and its value at 2 is 5.
Therefore, f is continuous at x = 2
Solution :
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.
Solution :
The given function f is
Therefore, f is continuous at all points x, such that x < −3
Case II:
Therefore, f is continuous at x = −3
Case III:
Therefore, f is continuous in (−3, 3).
Case IV:
If c = 3, then the left hand limit of f at x = 3 is,
The right hand limit of f at x = 3 is,
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:
Therefore, f is continuous at all points x, such that x > 3
Hence, x = 3 is the only point of discontinuity of f.
Solution :
Solution :
Solution :
Therefore,
f
is continuous at all points
x
, such that
x
> 1
Hence, the given function
f
has no point of discontinuity.
Solution :
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.
Solution :
The given function f is
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.
a continuous function?
Solution :
The given function is
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.
Solution :
The given function is f =
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
Solution :
The given function is
Question
16. Discuss the continuity of the function f, where f is defined by
Solution :
The given function f is
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
is continuous at x = 3.
Solution :
The given function f is
Question 18. For what value of λ is the function defined by
continuous at x = 0?
What about continuity at x = 1?
Solution :
The given function f is
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.
Solution :
The given function is
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 = π ?
Solution :
The given function is
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
Solution :
It is known that if g and h are two continuous functions, then
g + h, g - h and g.h are also continuous.
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
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
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,
Solution :
It is known that if g and h are two continuous functions, then
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 → c, then h → 0
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
Therefore, h (x) = cos x is continuous function.
It can be concluded that,
Question
23. Find the points of discontinuity of f, where
Solution :
The given function f is
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
is a continuous function?
Solution :
The given function f is
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
25. Examine the continuity of f, where f is defined by
Solution :
The given function f is
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.
Solution :
The given function f is
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.
Solution :
The given function is
Therefore, the required value of k is 3/4.
Question
28. Find the values of k so that the function f is continuous at the indicated point.
Solution :
The given function is
Therefore, the required value of k is -2/π
Question
29. Find the values of k so that the function f is continuous at the indicated point.
Solution :
The given function f is
Therefore, the required value of k is 9/5
Question
30. Find the values of a and b such that the function defined by
is a continuous function.
Solution :
The given function f is
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 (x
2
) is a continuous function.
Solution :
The given function is f (x) = cos (x
2
)
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) = x
2
It has to be first proved that g (x) = cos x and h (x) = x
2
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
Therefore, g (x) = cos x is continuous function.
h (x) = x
2
Clearly, h is defined for every real number.
Let k be a real number, then h (k) = k
2
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, h is a continuous function.
Question
32. Show that the function defined by f(x) = |cos x| is a continuous function.
Solution :
The given function is f(x) = |cos x|
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) = |x| and h(x) = cos x
It has to be first proved that g(x) = |x| and h(x) = cos x are continuous functions.
Clearly, g is defined for all real numbers.
Let c be a real number.
Case I:
Therefore, g is continuous at all points x, such that x < 0
Case II:
Therefore, g is continuous at all points x, such that x > 0
Case III:
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
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.
Question
33. Examine that sin|x| is a continuous function.
Solution :
Let, f(x) = sin|x|
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) = |x| and h (x) = sin x
It has to be proved first that g (x) = |x| and h (x) = sin x are continuous functions.
Clearly, g is defined for all real numbers.
Let c be a real number.
Case I:
.png)
Therefore, g is continuous at all points x, such that x < 0
Case II:
.png)
Therefore, g is continuous at all points x, such that x > 0
Case III:
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
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.
Question
34. Find all the points of discontinuity of f defined by
f(x) = |x| - |x + 1|.
Solution :
The given function is f(x) = |x| - |x + 1|
The two functions, g and h, are defined as
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.
