Let $\displaystyle f: [a,b] \rightarrow [a,b]$ be a continuous function. Prove that there is at least one point $\displaystyle x \in [a,b]$ so that $\displaystyle f(x)=x$.

This looks a lot like Intermediate Value Theorem but with the condition that $\displaystyle f(x)=x$. The condition that $\displaystyle f(x)=x$ is tricky for me, that there is an $\displaystyle x \in [a,b]$ that satisfies this. Thanks for any help.