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

• all the points of the open interval $I = (a,b)$ are shadow points;

• $a$ and $b$ are not shadow points.

Prove that

a) $f(x) \le f(b)$ for all $a < x < b$;

b) $f(a) = f(b)$.