CIIM 2015 Problem 4

Let $f:\mathbb{R} \to \mathbb{R}$ a continuos function and $\alpha$ a real number such that $$\lim_{x\to\infty}f(x) = \lim_{x\to-\infty}f(x) = \alpha.$$ Prove that for any $r > 0,$ there exists $x,y \in \mathbb{R}$ such that $y-x = r$ and $f(x) = f(y).$