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

It is given function is f(x) = cos (x²)

This function f is defined for every real number and f can be written as the composition of two function as,

f = goh, where, g(x) = cosx and h(x) = x²

First we have to prove that g(x) = cosx and h(x) = x² are continuous functions.

We know that g is defined for every real number.

Let k be a real number.

Then, g(k) =cos k

Now, put x = k + h

If

= coskcos0 – sinksin0

= cosk × 1 – sin × 0

= cosk

Thus, g(x) = cosx is continuous function.

Now, h(x) = x²

So, h is defined for every real number.

Let c be a real number, then h(c) = c²

Therefore, h is a continuous function.

We know that for real valued functions g and h,

Such that (fog) is continuous at c.

Therefore, f(x) = (goh)(x) = cos(x²) is a continuous function.