Single Variable Calculus, Chapter 2, 2.5, Section 2.5, Problem 27

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

We can rewrite
$\quad F(x) = g(x) + h(x)$ where $g(x) = \sqrt{x}$ and $h(x) = \sin x$
Both of the functions $g(x) = \sqrt{x}$ and $h(x) = \sin x$ are examples of the functions that are continuous on every number on its domain. $h(x) = \sin x$
has a domain of $(-\infty, \infty)$.

However, $g(x) = \sqrt{x}$ is defined only for its domain $[0, \infty)$,

Therefore,
$\quad$ The domain of $F(x)$ is $[0, \infty)$