Real Analysis Exchange

The composition of two connected G΄ functions has a fixed point.

Piotr Szuca

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We show that if $f,g\colon[0,1]\to[0,1]$ are both functions with connected $G_\delta$ graphs, then their composition has a fixed point. This is a generalization of the analogous result for Darboux Baire~1 functions.

Article information

Real Anal. Exchange, Volume 29, Number 2 (2003), 931-938.

First available in Project Euclid: 7 June 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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}
Secondary: 26A21: Classification of real functions; Baire classification of sets and functions [See also 03E15, 28A05, 54C50, 54H05]

Baire 1 functions Darboux functions connectivity functions compositions of functions fixed points.


Szuca, Piotr. The composition of two connected G ΄ functions has a fixed point. Real Anal. Exchange 29 (2003), no. 2, 931--938.

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