Functions for which all points are local extrema. (English) Zbl 1170.26002

Behrends, Ehrhard; Geschke, Stefan; Natkaniec, Tomasz

Real Anal. Exch. 33(2007-2008), No. 2, 467-470 (2008).

The authors give two elementary proofs that every continuous function \(f: [0,1]\to \mathbb R\) which has local maximum or minimum at each point is a constant. The first proof focuses on the topological properties of the set \([0,1]\). The general result is this:
Theorem 1. Let \(X\) be a connected separable metric space. Then every continuous function \(f: X\to\mathbb R\) with local maximum or minimum at each point is a constant. The second proof uses the linear order.
Theorem 2. Let \(X\) be a connected separable linearly ordered space. Then every continuous function \(f: X\to\mathbb R\) that has local extremum at each point is a constant. At the same time the authors provide an example of a compact connected linearly ordered space \(X\) and a continuous function \(f: X\to\mathbb R\) which has local extremum at each point, but is not constant.

Reviewer: Khristo N. Boyadzhiev (Ada)

MSC:

26A15	Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
54C30	Real-valued functions in general topology

Keywords:

metric space; separable space; local extremum; continuous function