IMC 2011 Problem 1

Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function. A point $x$ is called a shadow point if there exists a point $y\in \mathbb{R}$ with $y>x$ such that $f(y)>f(x).$ Let $a<b$ be real numbers and suppose that $\bullet$ all the points of the open interval $I=(a,b)$ are shadow points; $\bullet$ $a$ and $b$ are not shadow points. Prove that a) $f(x)\leq f(b)$ for all $a<x<b;$ b) $f(a)=f(b).$ Proposed by José Luis Díaz-Barrero, Barcelona