Chapter 5 Continuity and Differentiability Ex 5.1 Solutions
Question - 21 : - Discuss the continuity of the followingfunctions.
(a) f (x)= sin x + cos x
(b) f (x)= sin x − cos x
(c) f (x)= sin x × cos x
Answer - 21 : -
It isknown that if g and h are two continuousfunctions, then
are also continuous.
Ithas to proved first that g (x) = sin x and h (x)= cos x are continuous functions.
Let g (x)= sin x
It isevident that g (x) = sin x is definedfor every real number.
Let c bea real number. Put x = c + h
If x → c,then h → 0
Therefore, g isa continuous function.
Let h (x)= cos x
It isevident that h (x) = cos x is definedfor every real number.
Let c bea real number. Put x = c + h
If x → c,then h → 0
h (c) =cos c
Therefore, h isa 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 thecontinuity of the cosine, cosecant, secant and cotangent functions,
Answer - 22 : -
It isknown that if g and h are two continuousfunctions, then
Ithas to be proved first that g (x) = sin x and h (x)= cos x are continuous functions.
Let g (x)= sin x
It isevident that g (x) = sin x is definedfor every real number.
Let c bea real number. Put x = c + h
If x
c, then h
0
Therefore, g isa continuous function.
Let h (x)= cos x
It isevident that h (x) = cos x is definedfor every real number.
Let c bea 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 concludedthat,
Therefore, cosecant is continuous except at x = np, n Z
Therefore, secant is continuous except at 
Therefore,cotangent is continuous except at x = np, n Î Z
Question - 23 : - Findthe points of discontinuity of f, where
Answer - 23 : - The given function f is
It isevident that f is defined at all points of the real line.
Let c bea real number.
Case I:
Therefore, f iscontinuous at all points x, such that x < 0
Case II:
Therefore, f iscontinuous at all points x, such that x > 0
Case III:
Theleft hand limit of f at x = 0 is,
Theright hand limit of f at x = 0 is,
Therefore, f iscontinuous at x = 0
Fromthe above observations, it can be concluded that f iscontinuous at all points of the real line.
Thus, f hasno point of discontinuity.
Determineif f defined by
is a continuousfunction?
Answer - 24 : - The given function f is
It isevident that f is defined at all points of the real line.
Let c bea real number.
Case I:
Therefore, f iscontinuous at all points x ≠ 0
Case II:
Therefore, f iscontinuous at x = 0
Fromthe above observations, it can be concluded that f is continuousat every point of the real line.
Thus, f isa continuous function.
Examinethe continuity of f, where f is defined by
Answer - 25 : - The given function f is
It isevident that f is defined at all points of the real line.
Let c bea real number.
Case I:
Therefore, f iscontinuous at all points x, such that x ≠ 0
Case II:
Therefore, f iscontinuous at x = 0
Fromthe above observations, it can be concluded that f iscontinuous at every point of the real line.
Thus, f isa continuous function.
Find the values of k so that the function f is continuous at the indicated point.
Answer - 26 : - The given function f is
The given function f is continuous at
, if f is defined at and if the value of the f at equals the limit of f at It is evidentthat f is defined at and 
Therefore,the required value of k is 6.
Findthe values of k so that the function f iscontinuous at the indicated point.
Answer - 27 : - The given function is
Thegiven function f is continuous at x = 2,if f is defined at x = 2 and if the valueof f at x = 2 equals the limit of f at x =2
It is evidentthat f is defined at x = 2 and
Therefore, the required value of 
Question - 28 : - Findthe values of k so that the function f iscontinuous at the indicated point.
Answer - 28 : - The given function is
Thegiven function f is continuous at x = p,if f is defined at x = p and if the valueof f at x = p equals the limit of f at x =p
It is evidentthat f is defined at x = p and
Therefore, the required value of
Question - 29 : - Findthe values of k so that the function f iscontinuous at the indicated point.
Answer - 29 : - The given function f is
Thegiven function f is continuous at x = 5,if f is defined at x = 5 and if the valueof f at x = 5 equals the limit of f at x =5
It is evidentthat f is defined at x = 5 and
Therefore, the required value of
Question - 30 : - Findthe values of a and b such that the functiondefined by
is a continuousfunction.
Answer - 30 : - The given function f is
It isevident that the given function f is defined at all points ofthe real line.
If f isa continuous function, then f is continuous at all realnumbers.
Inparticular, f is continuous at x = 2and x = 10
Since f iscontinuous at x = 2, we obtainSince f iscontinuous at x = 10, we obtain
On subtracting equation (1) from equation(2), we obtain
8a =16
⇒ a = 2
Byputting 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 isa continuous function are 2 and 1 respectively.