Single Variable Calculus, Chapter 2, 2.5, Section 2.5, Problem 25

Explain using theorems of continuity why the function $h(x) = \cos ( 1 - x^2)$ is continuous at every number in its domain. State the domain


We can rewrite,
$\quad h(x) = f(g(x))$

Where,
$\quad f(x) = \cos x \text{ and } g(x) = 1-x^2$


The functions $f(x) = \cos x$ and $g(x) = 1 - x^2$ are examples of the functions that are continuous on every number in its domain according to the definition.
Also, from the definition, the composite function $h(x)$ will be continuous on every number on its domain as well.

Therefore,

$\quad $The domain of $h(x)$ is $(-\infty, \infty)$