Characterizing Distances Between Points in the Level Sets of a Class of Continuous Functions on a Closed Interval

Given a continuous function $f:[a,b]\to \mathbb{R}$ such that $f(a)=f(b)$, we investigate the set of distances $\mathrm{\mid }x-y\mathrm{\mid }$ where $f(x)=f(y)$. In particular, we show that the only distances this set must contain are ones which evenly divide $[a,b]$. Additionally, we show that it must contain at least one third of the interval $[0,b-a]$. Lastly, we explore some higher dimensional generalizations.