#### Question

Show that the function defined by* f *(*x*) = cos (*x*^{2}) is a continuous function.

#### Solution

The given function is *f *(*x*) = cos (*x*^{2})

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*) = *x*^{2}

`[∴(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*) = *x*^{2} 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*

*Put x = c + h*

Therefore, *h* is a continuous function.

It is known that for real valued functions *g *and *h*,such that (*g *o *h*) is defined at *c*, if *g *is continuous at *c *and if *f *is continuous at *g *(*c*), then (*f *o *g*) is continuous at *c*.

Therefore, `f(x) = (goh)(x) = cos(x^2)` is a continuous function.