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

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.

### 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
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