Points at Which Continuous Functions Have the Same Height

Abstract

As an easy application of the intermediate value theorem, one can show that for any continuous function f: [0, 1] → ℝ with f (0) = f (1), there are points a, a + 1/2 both in [0, 1] such that f (a) = f (a + 1/2). In this note, we show that this property holds with 1/2 replaced by any number of the form 1/n for a positive integer n. More interestingly, we show that this is false for every number not of the form 1/n.