Question

Show that the function defined by f(x)=|cosx| is a continuous function.

Answer 1

The given function is f(x)=|cosx|
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)=|x| and h(x)=cosx[∵(goh)(x)=g(h(x))=g(cosx)=|cosx|=f(x)]
It has to be first proved that g(x)=|x| and h(x)=cosx are continuous functions.
g(x)=|x| can be written as 
\(f(x)=\left\{\begin{matrix} -x, &if\,x<0 \\   x,&if\,x\geq0  \end{matrix}\right.\)
Clearly,g is defined for all real numbers.
Let c be a real number.

Case I:
If c<0,then g(c)=-c and \(\lim_{x\rightarrow c}\) g(x)=\(\lim_{x\rightarrow c}\)(-x)=-c
∴\(\lim_{x\rightarrow c}\)g(x)=g(c)
Therefore,g is continuous at all points x,such that x<0

Case II:
If c>0,then g(c)=c and \(\lim_{x\rightarrow c}\) g(x)=\(\lim_{x\rightarrow c}\)x=c
∴\(\lim_{x\rightarrow c}\)g(x)=g(c)
Therefore,g is continuous at all points x, such that x>0

Case III:
If c=0,then g(c)=g(0)=0
\(\lim_{h\rightarrow 0^-}\) g(x)=\(\lim_{h\rightarrow 0^-}\)(-x)=0
\(\lim_{x\rightarrow 0^+}\)= g(x)=\(\lim_{x\rightarrow 0^+}\)(x)=0
∴\(\lim_{x\rightarrow 0^-}\)g(x)=\(\lim_{x\rightarrow 0^+}\)(x)=g(0)
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)=cosx is defined for every real number.
Let c be a real number. 
Put x=c+h If x\(\rightarrow\)c, then h\(\rightarrow\)0 
h(c)=cosc
\(\lim_{x\rightarrow c}\)h(x)=\(\lim_{x\rightarrow c}\) cosx
=\(\lim_{x\rightarrow 0}\)cos(c+h)
=\(\lim_{x\rightarrow 0}\)[cos c cos h-sin c sin h]
=\(\lim_{x\rightarrow 0}\)cos c cos 0-sin c sin 0
=cos c cos 0-sin c sin 0
=cos c\(\times\)1-sinc\(\times\)0
=cos c
∴\(\lim_{x\rightarrow c}\)h(x)=h(c)

Therefore,h(x)=cos x 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)=g(h(x))=g(cosx)=|cosx|is a continuous function.