RO-NMO 2007 Problem 112

Let $f: \mathbb{R}\to\mathbb{R}$ be a continuous function, and $a<b$ be two points in the image of $f$ (that is, there exists $x,y$ such that $f(x)=a$ and $f(y)=b$ ).

Show that there is an interval $I$ such that $f(I)=[a,b]$ .