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Question 1:

## Prove that the function is continuous at

Therefore, f is continuous at x = 0

Therefore, f is continuous at x = −3

Therefore, f is continuous at x = 5

Question 2:

## Examine the continuity of the function .

Thus, f is continuous at x = 3

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Question 3:
Examine the following functions for continuity.

(a)
(b)

(c)
(d)

## (a) The given function is

It is evident that f is defined at every real number k and its value at k is k − 5.

## (b) The given function is

For any real number k ≠ 5, we obtain

function.

## (c) The given function is

For any real number c ≠ −5, we obtain

function.

## (d) The given function is

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

## Therefore, f is continuous at all real numbers less than 5.

Case II : c = 5

Then,

Therefore, f is continuous at x = 5
Case III: c > 5

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.

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,

## 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

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

Case (ii) c > 2

## 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,

## The right hand limit of f at x = 2 is,

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 < −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

## The given function f is

It is known that,
Therefore, the given function can be rewritten as

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 < 0

Case II:
If c = 0, then the left hand limit of f at x = 0 is,

## The right hand limit of f at x = 0 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

It is known that,

## Let c be any real number. Then,

Also,
Therefore, the given function is a continuous function.
Hence, the given function has no point of discontinuity.

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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:

Case II:

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

Therefore, f is continuous at x = 1
Case III:

## Therefore, f is continuous at all points x, such that x > 1

Hence, the given function f has no point of discontinuity.

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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

Case II:

Therefore, f is continuous at x = 2
Case III:

## 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.

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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

Case II:
If c = 1, then the left hand limit of f at x = 1 is,

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

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:

## 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.

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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:

Case II:

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

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:

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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:

Case II:

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

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:

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Case IV:

## The right hand limit of f at x = 3 is,

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:

## 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:

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Case II:

## The right hand limit of f at x = 0 is,

Therefore, f is continuous at x = 0
Case III:

Case IV:

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

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:

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## 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:

Case II:

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

Therefore, f is continuous at x = −1
Case III:

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Case IV:

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

Therefore, f is continuous at x = 2
Case V:

## 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

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## Therefore, the required relationship is given by,

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

## The given function f is

If f is continuous at x = 0, then

## 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

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Question 19:

## The given function is

It is evident that g is defined at all integral points.
Let n be an integer.
Then,

## The right hand limit of f at x = n 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:

## The given function is

It is evident that f is defined at x = p

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## 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
It is known that if g and h are two continuous functions, then

## 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

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## 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,

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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

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## Therefore, h (x) = cos x is continuous function.

It can be concluded that,

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

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

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## 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:

Case II:

Case III:

## The right hand limit of f at x = 0 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.

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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:

Case II:

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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:

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## Therefore, f is continuous at all points x, such that x ≠ 0

Case II:

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.

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## 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

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## Therefore, the required value of .

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

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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

## Therefore, the required value of

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

is a continuous function.

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## 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

## 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.

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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.

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Question 32:

## 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,

f = g o h, where

## Clearly, g is defined for all real numbers.

Let c be a real number.
Case I:

Case II:

Case III:

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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.

Question 33:

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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

## Clearly, g is defined for all real numbers.

Let c be a real number.
Case I:

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

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## 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.

Question 34:

## The given function is

The two functions, g and h, are defined as

Then, f = g − h
The continuity of g and h is examined first.

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## Clearly, g is defined for all real numbers.

Let c be a real number.
Case I:

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.

## Clearly, h is defined for every real number.

Let c be a real number.
Case I:

Case II:

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## Therefore, h is continuous at all points x, such that x > −1

Case III:

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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