Universal chord theorem

Let $\delta >0, f \in C(\left[0,1\right])$. We say that the graph of $f$ has a horizontal chord of length $\delta$ if there is a point $x \in [0,1-\delta ]$ such that $f(x)=f(x+\delta )$. Prove that if $f(0)=f(1)$, then for any $n \in \mathbb {N}$ there is a horizontal chord of length $\frac{1}{n}$. Show that for chords of other lengths this is, in general, false.