Answer
If the product function $h(x)=f(x)\times g(x)$ is continuous at $x=0$, it is still possible that either $f(x)$ or $g(x)$ is not continuous at $x=0$.
Work Step by Step
If the product function $h(x)=f(x)\times g(x)$ is continuous at $x=0$, it is still possible that either $f(x)$ or $g(x)$ is not continuous at $x=0$.
Consider this example: $f(x)=x$ and $g(x)=1/x$.
We see that here $g(x)$ is undefined at $x=0$, causing $g(x)$ to be discontinuous at $x=0$.
However, consider the product function $h(x)$: $$h(x)=f(x)\times g(x)=x\times\frac{1}{x}=1$$
At $x=0$: $\lim_{x\to0}h(x)=1=h(0)$. So $h(x)$ is still continuous at $x=0$, even though one of the function, here $g(x)$, is not continuous at $x=0$.
