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# NCERT Solutions - Continuity & Differentiability, Exercise 5.1 JEE Notes | EduRev

## JEE : NCERT Solutions - Continuity & Differentiability, Exercise 5.1 JEE Notes | EduRev

The document NCERT Solutions - Continuity & Differentiability, Exercise 5.1 JEE Notes | EduRev is a part of the JEE Course Mathematics (Maths) Class 12.
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Continuity & Differentiability

Question 1:
Prove that the function

Therefore, f is continuous at x = 5

Question 2:
Examine the continuity of the function
.

Thus, f is continuous at x = 3

Question 3: Examine the following functions for continuity.
(a)

(b)
(c)

(d)

(c) The given function is
For any real number c â‰  âˆ’5, we obtain

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.
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?
The given function f is
At x = 0,
It is evident that f is defined at 0 and its value at 0 is 0.

The right hand limit of f at x = 1 is,

Therefore, f is continuous at x = 2

Question 6: Find all points of discontinuity of f, where f is defined by

The given function f is
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

Therefore, f is continuous at all points x, such that x < 2
Case (ii) c > 2

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

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 < âˆ’3
Case II:

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

The given function f is

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:

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

The given function f is

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

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 > 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

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 > 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

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 > 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?
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.

Question 14: 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 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 (1, 3).
Case IV:

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

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:

Therefore, f is continuous at all points of the interval (0, 1).
Case IV:

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

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 x = âˆ’1

Case III:

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.

The given function f is
If f is continuous at x = 3, then

Question 18: For what value of is the function defined by

continuous at x = 0? What about continuity at x = 1?
The given function f is
If f is continuous at x = 0, then

Therefore, there is no value of Î» for which f is continuous at x = 0 At x = 1,
f (1) = 4x + 1 = 4 Ã— 1 + 1 = 5

Therefore, for any values of Î», f is continuous at x = 1

Question 19: Show that the function defined by g(x)= x-[x] is discontinuous at all integral point.
Here [x] denotes the greatest integer less than or equal to x.

The given function is g(x)= x-[x]
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 = p?
The given function is
It is evident that f is defined at x = p

Therefore, the given function f is continuous at x = Ï€

Question 21: Discuss the continuity of the following functions.
(a) f (x) = sin x + cos x
(b) f (x) = sin x âˆ’ cos x
(c) f (x) = sin x Ã— 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 â†’ c, then h â†’ 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 â†’ c, then h â†’ 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) âˆ’ h (x) = sin x âˆ’ cos x is a continuous function
(c) f (x) = g (x) Ã— h (x) = sin x Ã— cos x is a continuous function.

Question 22: 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

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 â†’ c, then h â†’ 0
Therefore, h (x) = cos x is continuous function.

Therefore, cotangent is continuous except at x = np, n ÃŽ Z

Question 23: Find the points of discontinuity of f, where

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?
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

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.

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.

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

Question 28: 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

Question 29: 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

Question 30: Find the values of a and b such that the function defined by

, is a continuous function.
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.
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

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,   is a continuous function.

Question 32: Show that the function defined by  is a continuous function.
The given function is
This function f is defined for every real number and f can be written as the composition of two functions as,

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.

Question 33: Examine that sin|x| is a continuous function.

This function f is defined for every real number and f can be written as the composition
of two functions as,

Clearly, g is defined for all real numbers.
Let c be a real number.
Case I:

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 â†’ c, then k â†’ 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.

is a continuous function.

Question 34:
Find all the points of discontinuity of f defined by
.

The given function is
The two functions, g and h, are defined as

Clearly, g is defined for all real numbers.
Let c be a real number.

Case I:

From the above three observations, it can be concluded that g is continuous at all

Case II:

Clearly, h is defined for every real number.
Let c be a real number.
Case I:

Therefore, h is continuous at all points x, such that x < âˆ’1
Case II:

Therefore, h is continuous at x = âˆ’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 âˆ’ h is also acontinuous function.
Therefore, f has no point of discontinuity.

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