Real Analysis Exchange

Functions for Which All Points Are Local Extrema

Ehrhard Behrends, Stefan Geschke, and Tomasz Natkaniec

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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 $f$.

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Real Anal. Exchange Volume 33, Number 2 (2007), 467-470.

First available in Project Euclid: 18 December 2008

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Mathematical Reviews number (MathSciNet)

Primary: 26A15: Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) {For properties determined by Fourier coefficients, see 42A16; for those determined by approximation properties, see 41A25, 41A27} 54C30: Real-valued functions [See also 26-XX]

local extremum continuous function


Behrends, Ehrhard; Geschke, Stefan; Natkaniec, Tomasz. Functions for Which All Points Are Local Extrema. Real Anal. Exchange 33 (2007), no. 2, 467--470.

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  • M. R. Wojcik, problem session, $34^{th}$ Winter School in Abstract Analysis, Lhota nad Rohanovem, Czech Republik (2006).