Let $f:\mathbb{R}\rightarrow
\mathbb{R}$ and
$g:\mathbb{R}\rightarrow \mathbb{R}$
be continuous. Is
$h:\mathbb{R}\rightarrow
\mathbb{R}$, where $h(x): = f(x)
\times g(x)$, still continuous?
I guess it is, but I feel difficult to manipulate the absolute difference:
$$|h(x_2)-h(x_1)|=|f(x_2)g(x_2)-f(x_1)g(x_1)| \dots $$
Thanks in advance!