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

Idea: Such problems can be solved easily using the idea of the continuity of composite function.

If we not go to very strict mathematical meaning of composite function you can think it as it is a function of function.

Let, g(x) = cos x

And h(x) = x²

Then g(h(x)) = g(x²) = cos x²

We write g(h(x)) as (goh)(x) and this is what we called composite function/function composition.

We have a theorem regarding composition of function in continuity which lets us to solve problems easily.

Theorem: If f and g are real valued function such that (fog) is defined at c, and g is continuous at c and f is continuous at g(c) then (fog) is continuous at x = c

For our problem:

Let, g(x) = cos x

and h(x) = x²

Given: f(x) = cos x² = g(h(x)) = (goh)(x)

Clearly, h(x) is a polynomial function, which is everywhere continuous

And g(x) being cosine function, it is also everywhere continuous.

FIG : Plot of cos x²

∴ goh(x) = f(x) is also everywhere continuous. [using above explained theorem]