This is a problem that my professor has in his notes. I could really use some guidance. I don't understand how to prove problems such as these.

Let $\displaystyle f:[a,b]\rightarrow R$ be continuous. If there is a $\displaystyle c<1$, such that for any$\displaystyle x\in [a,b], \exists y \in [a,b], |f(y)|\leq c|f(x)|$, then prove that there is a $\displaystyle r\in [a,b]$ such that $\displaystyle f(r)=0$. [Hint: Consider g(x)=|f(x)|]