Let $(X,d)$ be a compact metric space. Let $f: X \to \mathbb{R}$ be a continuous function such that for each $x \in X$ there is a $y \in X$ with $|f(x)| \leq \frac{1}{3}|f(y)|$. Show that there is a point $c \in X$ such that $f(x)=0$.
I'm stuck on this problem. At first I thought it was an application of the Intermediate Value Theorem. I don't know if that is the case.