Proof Writing & Problem Solving · Superchallenge problem

Let $f$ be continuous on $[0, 1]$ with $f(0) = f(1)$. A horizontal chord of length $h$ means some $x$ with $0 \le x \le 1 - h$ and $f(x + h) = f(x)$. For how many of the nine lengths $h = 0.1, 0.2, \ldots, 0.9$ is a horizontal chord of length $h$ GUARANTEED for every such $f$?