If $f\cdot g$ is continuous at $x=a$, then are $f$ and $g$ both continuous at $x=a$?

Question

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?

Answer 1

It is false. As example You can take $f=x$, $g=\theta(x)$ (Heaviside function)

Answer 2

Take $f = 1_{[a,\infty)}$ and $g = 1_{(-\infty,a)}$. Then $fg = 0$, but none are continuous at $a$.