On locally extremal functions on connected spaces

TL;DR

This paper constructs a continuous, non-constant function on a connected complete metric space where every point is a local extremum, illustrating a unique topological property related to local extrema in non-separable spaces.

Contribution

It provides the first example of a connected, complete metric space with a continuous function where all points are local extrema, despite the space not being separably connected.

Findings

Existence of a continuous non-constant function with all points as local extrema

Connected complete metric space can lack separable connectedness

Illustrates topological distinctions in extremal functions

Abstract

We construct an example of a real-valued continuous non-constant function $f$ defined on a connected complete metric space $X$ such that every point of $X$ is a point of local minimum or local maximum for $f$. The space $X$ is connected but fails to be separably connected.