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Question
Show that the function defined by f(x) =|cosx| is continuous function.
Solution
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 = g o h, 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) = cos x are continous functions
g(x) = |x| can be written as
`g(x)={(-x,if x<0),(x,ifx>=0):}`
Clearly , g is defined for all real number
Let c be real number
Case I:
if c<0, then g(c)=-c and `lim_(x->c)g(x)=lim_(x->c)(-x)=-c`
`:.lim_(x->c)g(x)_=g(c)`
Therefore g is continous at all point x such that x < 0
Case II:
if c>0 then g(c)=c and `lim_(x->c)g(x)=lim_(x->c)x=c`
`:.lim_(x->c)g(x)=g(c)`
Therefore g is continous at all point x such that x > 0
Case III:
if c = 0, then g(c)= g(0)=0
`lim_(x->0^-)g(x)=lim_(x->0^-)(-x)=0`
`lim_(x->0^+)g(x)=lim_(x->0^+)(x)=0`
`:.lim_(x->0^+)g(x)=lim_(x->0^+)(x)=g(0)`
There g is continous at x = 0
From the above three observation, it can be concluded that g is continous at all point
h(x) = cosx
It is evident that h(x)=cosx is defined for every real number.
Let c be ral number Put x= c+h
if x → c,then h → 0
h(c) = cos c
`lim_(x->c)h(x)=lim_(x->c)cosx`
`=lim_(h->0)cos (c+h)`
`=lim_(h->0)[cosc cosh-sinc sinh]`
`=lim_(h->0)cosc cosh-lim_(h->0)sinc sinh`
= cos c cos0 - sinc sin0
= cos c x 1 - sin c x 0
= cos c
`:.lim_(x->c)h(x)=h(c)`
Therefore h(x)= cosx is continuous function
It is known that for real valued functions g and h,m such that (goh) is defined at c, if g is continuous at c and if f is continous at g(c) then (fog) is continous at c.
Therefore, f(x)=(goh)(x)=g(h(x)) = g(cosx)= |cosx| is a continuous function
