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!