NCERT Solutions for Class 12 Maths Chapter 5: Continuity and Differentiability
Continuity and Differentiability is a fundamental chapter in the Class 12 Maths syllabus, offering a comprehensive understanding of essential calculus concepts. Chapter 5 delves into the continuity of functions, differentiability, and their applications, forming the foundation for advanced mathematical analysis. For students of Class 12, mastering this chapter is crucial for excelling in board exams and competitive tests.
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The NCERT Solutions for Class 12 Maths Chapter 5: Continuity and Differentiability are tailored to help the students master the concepts that are key to success in their classrooms. The solutions given in the PDF are developed by experts and correlate with the CBSE syllabus of 2023-2024. These solutions provide thorough explanations with a step-by-step approach to solving problems. Students can easily get a hold of the subject and learn the basics with a deeper understanding. Additionally, they can practice better, be confident, and perform well in their examinations with the support of this PDF.
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Access Answers to NCERT Solutions for Class 12 Maths Chapter 5: Continuity and Differentiability
Students can access the NCERT Solutions for Class 12 Maths Chapter 5: Continuity and Differentiability. Curated by experts according to the CBSE syllabus for 2023–2024, these step-by-step solutions make Maths much easier to understand and learn for the students. These solutions can be used in practice by students to attain skills in solving problems, reinforce important learning objectives, and be well-prepared for tests.
Continuity and Differentiability : Exercise 5.1
Prove that the function f(x) = 5x – 3 is continuous at x = 0, at x = – 3 and x = 5
Exercise 5.2
Examine the continuity of the function f(x) = 2x2 – 1 at x = 3
Thus, f is continuous at x = 3
Exercise 5.3
Examine the following functions for continuity.
(a)
(c)
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.
Exercise 5.4
Prove that the function f(x) = xn is continuous at x = n, where n is a positive integer.
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.
Therefore, f is continuous at n, where n is a positive integer.
Exercise 5.5
Is the function f defined by
continuous at x = 0? At x = 1? At x = 2?
The given function f is
At x = 0,
It is evident that f is defined at 0 and its value at 0 is 0.
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
Exercise 5.6
Find all points of discontinuity of f, where f is defined by
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.
Exercise 5.7
Find all points of discontinuity of f, where f is defined by
The given function f is
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:
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.
Exercise 5.8
Find all points of discontinuity of f, where f is defined by
Exercise 5.9
Examine that sin|x| is a continuous function.
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:
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) = 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, 
Find all points of discontinuity of f, where f is defined by
Find all points of discontinuity of f, where f is defined by
Therefore, f is continuous at all points x, such that x > 1
Hence, the given function f has no point of discontinuity.
Find all points of discontinuity of f, where f is defined by
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.
Find all points of discontinuity of f, where f is defined by
The given function f is
The given function f is defined at all the points of the real line.
Let c be a point on the real line.
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.
Is the function defined by 
The given function is
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:
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.
Discuss the continuity of the function f, where f is defined by
f =
The given function is f =
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:
Therefore, f is continuous at all points of the interval (3, 10].
Hence, f is not continuous at x = 1 and x = 3
Discuss the continuity of the function f, where f is defined by
The given function is
The given function is defined at all points of the real line.
Let c be a point on the real line.
Case I:
Discuss the continuity of the function f, where f is defined by
The given function f is
The given function is defined at all points of the real line.
Let c be a point on the real line.
Case I:
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.
Find the relationship between a and b so that the function f defined by
The given function f is
If f is continuous at x = 3, then
For what value of λ is the function defined by
What about continuity at x = 1?
The given function f is
If f is continuous at x = 0, then
Therefore, for any values of λ, f is continuous at x = 1
Show that the function defined by 
The given function is
It is evident that g is defined at all integral points.
Let n be an integer.
Then,
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.
Is the function defined by 
The given function is
It is evident that f is defined at x = π
Therefore, the given function f is continuous at x = π
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
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
Discuss the continuity of the cosine, cosecant, secant and cotangent functions,
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,
Find the points of discontinuity of f, where
The given function f is
It is evident that f is defined at all points of the real line.
Let c be a real number.
Case I:
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.
Determine if f defined by 
The given function f is
It is evident that f is defined at all points of the real line.
Let c be a real number.
Case I:
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.
Examine the continuity of f, where f is defined by
The given function f is
It is evident that f is defined at all points of the real line.
Let c be a real number.
Case I:
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.
Find the values of k so that the function f is continuous at the indicated point.
The given function f is
The given function f is continuous at x = π/2 , if f is defined at x = π/2 and if the value of the f at x = π/2 equals the limit of f at x = π/2 .
It is evident that f is defined at x = π/2 and f( π/2) = 3
Therefore, the required value of k is 6.
Find the values of k so that the function f is continuous at the indicated point.
The given function is
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 at x = 2
It is evident that f is defined at x = 2 and f(2) = k(2)2 = 4k
Therefore, the required value of k is 3/4.
Find the values of k so that the function f is continuous at the indicated point.
The given function is
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 at x = p
It is evident that f is defined at x = p and f(π) = kπ + 1
Therefore, the required value of k is -2/π
Find the values of k so that the function f is continuous at the indicated point.
The given function f is
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 at x = 5
It is evident that f is defined at x = 5 and f(5) = kx + 1 = 5k + 1
Therefore, the required value of k is 9/5
Show that the function defined by f(x) = |cos x| is a continuous function.
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, 
Find the values of a and b such that the function defined by
The given function f is
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
Therefore, the values of a and b for which f is a continuous function are 2 and 1 respectively.
Show that the function defined by f (x) = cos (x2) is a continuous function.
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
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
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
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.
Find all the points of discontinuity of f defined by f(x) = |x| – |x + 1|.
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.
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