I'll state the question from my textbook below:
State whether the below statement is true or false:
If $f\cdot g$ is continuous at $x=a$, then $f$ and $g$ are both continuous at $x=a$.
The textbook says that it is false.
I tried thinking of a very basic example considering $f(x) = x$ and $g(x) = \frac 1x$. But then I recalled that $p(x) = \frac xx$ and $q(x) = 1$ are different functions. So, it didn't work as a counter example.
So, can someone please provide a counter example? Or is the statement true, just as I think?