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Question
Show that f (x) = cos x2 is a continuous function.
Solution
Given: 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 = g o h, where g (x) = cos x and h (x) = x2
`[∵ (goh)(x)=g(h (x))=g(x^2)=cos(x^2)=f(x)]`
It has to be first proved that g (x) = cos x 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) = cos c
`"If" x-> c , `then `h->0`
`lim_(x->c)g(x)=lim_(x->c)cos x`
`=lim_(h->0) cos (c+h)`
`=lim_(h->0)[cos c cos h-sin c sin h]`
`=lim_(h->0) cos c cos h -lim_(h->0) sin c sin h`
`=cos c cos 0-sin c sin 0`
`= cos cxx1-sin cxx0`
`= cos c`
`∴lim_(x->c)g(x)=g(c)`
So, g (x) = cos x is a continuous function.
Now,
h (x) = x2
Clearly, h is defined for every real number.
Let k be a real number, then h (k) = k2
`lim_(x->k)h(x)=lim_(x->k) x^2=k^2`
`∴lim_(x->k)h(x)=h(k)`
So, h is a continuous function.
It is known that for real valued functions g and h, such that (g o h) is defined at x = c, if g is continuous at x = c and if f is continuous at g (c), then, (f o g) is continuous at x= c.
Therefore, `f(x)=(goh)(x)=cos(x^2)`is a continuous function.
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