Question

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

Answer 1

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.