Question

Show that the function defined by f(x)=cos(x2) is a continuous function.

Answer 1

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=goh, where g(x)=cosx and h(x)=x2[∵(goh)(x)=g(h(x))=g(x2)=cos(x2)=f(x)]
It has to be first proved that g(x)=cosx 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)=cosc
put x=c+h
If x$\rightarrow$c,then h$\rightarrow$0
$\lim_{x\rightarrow c}$g(x)=$\lim_{x\rightarrow c}$ cosx
=$\lim_{h\rightarrow 0}$ cos(c+h)
=$\lim_{h\rightarrow 0}$[cos c cos h-sin c sin h]
=$\lim_{h\rightarrow 0}$ cos ccos 0-sin c sin 0
=cos c $\times$1-sin c $\times$0
=cos c
∴$\lim_{x\rightarrow c}$g(x)=g(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
$\lim_{x\rightarrow k}$h(x)=$\lim_{x\rightarrow k}$x2=k2
∴$\lim_{x\rightarrow k}$h(x)=h(k)
Therefore, h is a continuous function.
It is known that for real-valued functions g and h, such that (goh) is defined at c,if g is continuous at c and if f is continuous at g(c),then (fog) is continuous at c.
Therefore,f(x)=(goh)(x)=cos(x2) is a continuous function.