I do not know an example. But I will ask questions if in doubt of the example that you all may provide. Thank you!
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HINT: There are very simple functions $f(x)$ and $g(x)$ such that $f(x)$ and $g(x)$ are continuous everywhere except at $x=0$, and $f(x)g(x)=0$ for all $x\in\Bbb R$, so that $fg$ is continuous everywhere. You can choose $f(x)$ and $g(x)$ so that each of these functions takes on only the values $0$ and $1$.