Chapter 5 Continuity and Differentiability Ex 5.1 Solutions
Question - 31 : - Showthat the function defined by f (x) = cos (x2)is a continuous function.
Answer - 31 : -
Thegiven function is f (x) = cos (x2)
Thisfunction f is defined for every real number and f canbe written as the composition of two functions as,
f = g o h,where g (x) = cos x and h (x)= x2
Ithas to be first proved that g (x) = cos x and h (x)= x2 are continuous functions.
It isevident that g is defined for every real number.
Let c bea real number.
Then, g (c)= cos c
Therefore, g (x)= cos x is continuous function.
h (x)= x2
Clearly, h isdefined for every real number.
Let k bea real number, then h (k) = k2
Therefore, h isa continuous function.
It isknown that for real valued functions g and h,suchthat (g o h) is defined at c, if g iscontinuous at c and if f is continuousat g (c), then (f o g) iscontinuous at c.
Therefore, 
is a continuousfunction.
Question - 32 : - Show that the function defined by
is a continuous function.
Answer - 32 : - The given function is
Thisfunction f is defined for every real number and f canbe written as the composition of two functions as,
f = g o h,where
It has to be first proved that are continuous functions. Clearly, g isdefined for all real numbers.
Let c bea real number.
Case I:
Therefore, g iscontinuous at all points x, such that x < 0
Case II:
Therefore, g iscontinuous at all points x, such that x > 0
Case III:
Therefore, g iscontinuous at x = 0
Fromthe above three observations, it can be concluded that g iscontinuous at all points.
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 a continuous function.
It isknown that for real valued functions g and h,suchthat (g o h) is defined at c, if g iscontinuous at c and if f is continuousat g (c), then (f o g) iscontinuous at c.
Therefore, 
is a continuousfunction.
Question - 33 : - Examine that 
is a continuousfunction.
Answer - 33 : -
Thisfunction f is defined for every real number and f canbe written as the composition of two functions as,
f = g o h,where
It has to be proved first that are continuous functions. Clearly, g isdefined for all real numbers.
Let c bea real number.
Case I:
Therefore, g iscontinuous at all points x, such that x < 0
Case II:
Therefore, g iscontinuous at all points x, such that x > 0
Case III:
Therefore, g iscontinuous at x = 0
Fromthe above three observations, it can be concluded that g iscontinuous at all points.
h (x) =sin x
It isevident that h (x) = sin x is definedfor every real number.
Let c bea real number. Put x = c + k
If x → c,then k → 0
h (c) =sin c
Therefore, h isa continuous function.
It isknown that for real valued functions g and h,suchthat (g o h) is defined at c, if g iscontinuous at c and if f is continuousat g (c), then (f o g) iscontinuous at c.
Therefore, 
is a continuousfunction.
Question - 34 : - Find all the points of discontinuity of f defined by
Answer - 34 : - The given function is
Thetwo functions, g and h, are defined as
Then, f = g − h
Thecontinuity of g and h is examined first.
Clearly, g isdefined for all real numbers.
Let c bea real number.
Case I:
Therefore, g iscontinuous at all points x, such that x < 0
Case II:
Therefore, g iscontinuous at all points x, such that x > 0
Case III:
Therefore, g iscontinuous at x = 0
Fromthe above three observations, it can be concluded that g iscontinuous at all points.
Clearly, h isdefined for every real number.
Let c bea real number.
Case I:
Therefore, h iscontinuous at all points x, such that x < −1
Case II:
Therefore, h iscontinuous at all points x, such that x > −1
Case III:
Therefore, h iscontinuous at x = −1
Fromthe above three observations, it can be concluded that h iscontinuous at all points of the real line.
g and h arecontinuous functions. Therefore, f = g − h isalso a continuous function.
Therefore, f hasno point of discontinuity.