NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.1 – Continuity and Differentiability
Class 12 Maths Chapter 5 Exercise 5.1 – Continuity and Differentiability:- Class 12 Maths Chapter 5 Exercise 5.1 – Continuity and Differentiability introduces students to the fundamental concepts of continuity and differentiability in functions. This exercise focuses on understanding when and how a function is continuous, ensuring there are no abrupt changes in its value, and differentiable, meaning it has a defined tangent at every point.
By exploring these concepts, students gain a deeper insight into the behaviour of functions, which is crucial for advanced calculus and real-world applications. The exercise includes a variety of problems designed to reinforce these principles through practical examples and theorems. Check out the Class 12 Maths Chapter 5 Exercise 5.1 – Continuity and Differentiability NCERT Solutions from the below article.
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NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.1 – Continuity and Differentiability
Along with the Class 12 Maths Chapter 5 Exercise 5.1 – Continuity and Differentiability NCERT Solutions, check out the other Class 12 Maths Chapter 5 Exercise solutions:-
Question 1. Prove that the function f(x) = 5x – 3 is continuous at x = 0, at x = – 3 and x = 5
Solution :
Question 2. Examine the continuity of the function f(x) = 2x2 – 1 at x = 3
Solution :
Thus, f is continuous at x = 3
Question 3. Examine the following functions for continuity.
(a)
(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.
Question 4. Prove that the function f(x) = xn is continuous at x = n, where n is a positive integer.
Solution :
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.
Question 5. Is the function f defined by
continuous at x = 0? At x = 1? At x = 2?
Solution :
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
Check Out: NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.2
Question 6. Find all points of discontinuity of f, where f is defined by
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.
Question 7. Find all points of discontinuity of f, where f is defined by
Solution :
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.
Question 8. Find all points of discontinuity of f, where f is defined by
Solution :
Check out: NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.3
Question 9. Find all points of discontinuity of f, where f is defined by
Solution :
Question 10. Find all points of discontinuity of f, where f is defined by
Solution :
Therefore, f is continuous at all points x, such that x > 1
Hence, the given function f has no point of discontinuity.
Check out: NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.5
Question 11. Find all points of discontinuity of f, where f is defined by
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.
Check out: NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.6
Question 12. Find all points of discontinuity of f, where f is defined by
Solution :
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.
Question 13. Is the function defined by a continuous function?
Solution :
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.
NCERT Solutions for Class 12 Maths Chapter 5 Exercise 5.8
Question 14. Discuss the continuity of the function f, where f is defined by
f =
Solution :
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
Question 15. Discuss the continuity of the function f, where f is defined by
Solution :
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:
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Question 16. Discuss the continuity of the function f, where f is defined by
Solution :
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.
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
If f is continuous at x = 3, then
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
If f is continuous at x = 0, then
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 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.
Question 20. Is the function defined by continuous at x = π ?
Solution :
The given function is
It is evident that f is defined at x = π
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,
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Question 23. Find the points of discontinuity of f, where
Solution :
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.
Question 24. Determine if f defined by is a continuous function?
Solution :
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.
Question 25. Examine the continuity of f, where f is defined by
Solution :
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.
Question 26. Find the values of k so that the function f is continuous at the indicated point.
Solution :
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.
Question 27. Find the values of k so that the function f is continuous at the indicated point.
Solution :
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.
Question 28. Find the values of k so that the function f is continuous at the indicated point.
Solution :
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
Question 29. Find the values of k so that the function f is continuous at the indicated point.
Solution :
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
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
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.
Question 31. Show that the function defined by f (x) = cos (x2) is a continuous function.
Solution :
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.
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.
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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:
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, 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.
Class 12 Maths Chapter 5 Exercise 5.1 – Continuity and Differentiability Summary
This exercise introduces the important concepts of continuity and differentiability in functions. It starts by explaining what it means for a function to be continuous at a point and over an interval, highlighting the necessary conditions for continuity. It also covers differentiability, showing that a function is differentiable at a point if it has a well-defined tangent. The exercise also emphasises that a differentiable function is always continuous.
Continuity
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Definition and conditions for a function to be continuous.
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Methods to check the continuity of functions at points and over intervals.
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Continuity in composite functions.
Differentiability
-
Definition and conditions for differentiability.
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The relationship between continuity and differentiability.
-
Differentiability in composite functions.
Important Theorems
-
Intermediate Value Theorem.
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Rolle’s Theorem.
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Mean Value Theorem.
Class 12 Maths Chapter 5 Exercise 5.1 – Continuity and Differentiability FAQs
Q1. What is the definition of continuity at a point?
Ans. A function f(x)f(x)f(x) is continuous at a point x=cx = cx=c if limx→cf(x)=f(c)\lim_{x \to c} f(x) = f(c)limx→cf(x)=f(c).
Q2. What are the conditions for a function to be continuous?
Ans. A function is continuous at a point x=cx = cx=c if:
-
f(c)f(c)f(c) is defined.
-
limx→cf(x)\lim_{x \to c} f(x)limx→cf(x) exists.
-
limx→cf(x)=f(c)\lim_{x \to c} f(x) = f(c)limx→cf(x)=f(c).
Q3. How can we check the continuity of a function over an interval?
Ans. To check the continuity of a function over an interval, verify that the function is continuous at every point within the interval.
Q4. What is differentiability at a point?
Ans. A function f(x)f(x)f(x) is differentiable at a point x=cx = cx=c if the derivative f′(c)f'(c)f′(c) exists, which means limh→0f(c+h)−f(c)h\lim_{h \to 0} \frac{f(c+h) - f(c)}{h}limh→0hf(c+h)−f(c) exists.
Q5. Is every continuous function differentiable?
Ans. No, not every continuous function is differentiable. However, every differentiable function is continuous.