Answer
See the explanation below.
Work Step by Step
No.
Take
$f(x)=\left\{\begin{array}{lll}
0.5 & for & x\leq 0\\
1 & for & x\gt 0
\end{array}\right. $and $g(x)=\left\{\begin{array}{lll}
1 & for & x\leq 0\\
0.5 & for & x\gt 0
\end{array}\right..$
Neither is continuous at x=0 (the one-sided limits are different for each function), but the product is:
$h(x)=f(x)\cdot g(x)=\left\{\begin{array}{lll}
0.5 & for & x\leq 0\\
0.5 & for & x\gt 0
\end{array}\right.$
$h(x)=0.5$, for all x.
$h$ is a constant function, continuous everywhere, including at x=0..
