Continuity of $h(x)=f(x) \cdot g(x)$

Question

$h(x)=f(x) \cdot g(x)$

I want to check whether this function is continuous in its domain $\mathbb{R}$ or not.

definition by cases:

1. $f(x)$ and $g(x)$ are both continuous $\Rightarrow f(x) \cdot g(x)$ is continuous (Composition of continuous functions)

2. Let w.l.o.g g at $x_0$ be not continuous, this means $g(x) \neq g(x_0). \Rightarrow \lim\limits_{x \rightarrow x_0}((f \cdot g) (x)) = \lim\limits_{x \rightarrow x_0} f(x) \cdot \lim\limits_{x \rightarrow x_0} g(x) = f(x_0) \cdot \lim\limits_{x \rightarrow x_0} g(x) \neq f(x_0) \cdot g(x_0)$

$\Rightarrow f(x) \cdot g(x)$ is at the same points discontinuous like the function $g$

This is what i've got so far... Is this correct? Is it necessary to check what happens, if $f, g$ are both discontinuous?

Answer 1

The first statement in fully correct, yet for the second one you can find a really easy counterexample by taking $f(x)=0$.