If we let $\displaystyle I := [a,b]$ and let $\displaystyle f:I\rightarrow\mathbb{R}$ be a continuous function on $\displaystyle I$ such that for each $\displaystyle x$ in $\displaystyle I$ there exists $\displaystyle y$ in $\displaystyle I$ such that $\displaystyle |f(y)| \leq \frac{1}{2}|f(x)|$. Prove there exists a point $\displaystyle c$ in $\displaystyle I$ such that $\displaystyle f(c) = 0$.