Single Variable Calculus, Chapter 2, 2.5, Section 2.5, Problem 28

Explain using theorems of continuity why the function $F(x) = \sin(\cos(\sin x))$ is continuous at every number in its domain. State the domain


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

Where,
$\quad f(x) = \sin x, \quad g(x) = \cos x \quad \text{ and } \quad h(x) = \sin x$


The functions $f(x) = \sin x$ , $g(x) = \cos x$ and $h(x) = \sin x $ are all trigonometric functions that are continuous on every number in its domain according to the definition.
Also, from the definition, the composite function $F(x)$ will be continuous on every number on its domain as well.

Therefore,

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