Is product of two continuous functions still continuous? 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!

Hint: 
$$\left| f(x+h)g(x+h) - f(x)g(x) \right| = \left| f(x+h)\left( g(x+h) - g(x) \right) + \left( f(x+h) - f(x) \right) g(x) \right|$$
Can you proceed from here?
