Functions for Which All Points Are Local Extrema

Abstract

Let $X$ be a connected separable linear order, a connected separable metric space, or a connected, locally connected complete metric space. We show that every continuous function $f:X\to\mathbb R$ with the property that every $x\in X$ is a local maximum or minimum of $f$ is in fact constant. We provide an example of a compact connected linear order $X$ and a continuous function $f:X\to\mathbb R$ that is not constant and yet every point of $X$ is a local minimum or maximum of $f$.