real analysis - Proving continuity of a function using epsilon and delta

I've just got a real quick question about proving the continuity of a function using $\epsilon$ and $\delta$ definition of continuity. The question is this:

Let $f\colon X\to R$ be continuous where $X$ is some subset of $R$. Prove that the function $1/f\colon x\mapsto 1/f(x)$ is continuous at $p$ in $X$, provided that $f(p)\neq0$.

The definition states that "A function $f(x)$ is continuous at $p$ iff for every $\epsilon>0$ there exists some $\delta>0$ such that
$|x-p|<\delta$ and $|f(x) -f(p)|<\epsilon$

After that, I am super stuck...any help would be greatly appreciated.

Thanks!