Is it appropriate to consider a hole in the graph a zero?

Question

Let there be a function $$f(x) = \frac{x^2}{e^x-1}$$ which has a hole at $x=0$. It also approaches $f(x) = 0$ at this point. Would it be appropriate to call this hole a "zero" or not?

Answer 1

A zero is a point $a$ where $f(a) = 0$. In your case, $f(0)$ is undefined, so no, $x=0$ is not a zero.

On the other hand, since the discontinuity is removable, we can plug the hole by defining $$g(x)=\begin{cases} f(x), &x \neq 0\\ 0, &x=0 \end{cases}$$ in which case $x=0$ is indeed a zero of $g(x)$.