Real Analysis Exchange

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 $f$.

Article information

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

Dates
First available: 18 December 2008

http://projecteuclid.org/euclid.rae/1229619424

Mathematical Reviews number (MathSciNet)
MR2458263

Zentralblatt MATH identifier
05499484

Citation

Behrends, Ehrhard; Geschke, Stefan; Natkaniec, Tomasz. Functions for Which All Points Are Local Extrema. Real Analysis Exchange 33 (2007), no. 2, 467--470. http://projecteuclid.org/euclid.rae/1229619424.

References

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