## Real Analysis Exchange

- Real Anal. Exchange
- Volume 29, Number 1 (2003), 205-209.

### Loops of intervals and Darboux Baire 1 fixed point problem.

#### Abstract

WWe show that the existence of a loop of covering intervals for connected $G_\delta$ real functions implies the existence of a periodic point with its period equal to the length of the cycle. As a corollary we get that the composition of $N$ connected $G_\delta$ real functions from a compact interval into itself has a fixed point for every natural number $N$. In particular, the composition of finitely many derivatives from $[0,1]$ to $[0,1]$ has a fixed point. This solve the a problem from [3].

#### Article information

**Source**

Real Anal. Exchange, Volume 29, Number 1 (2003), 205-209.

**Dates**

First available in Project Euclid: 9 June 2006

**Permanent link to this document**

https://projecteuclid.org/euclid.rae/1149860185

**Mathematical Reviews number (MathSciNet)**

MR2061304

**Zentralblatt MATH identifier**

1065.26004

**Subjects**

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: 26A18: Iteration [See also 37Bxx, 37Cxx, 37Exx, 39B12, 47H10, 54H25] 26A21: Classification of real functions; Baire classification of sets and functions [See also 03E15, 28A05, 54C50, 54H05] 54C30: Real-valued functions [See also 26-XX] 37E99: None of the above, but in this section

**Keywords**

Derivatives connectivity functions Darboux functions Baire~1 functions Borel measurable functions composition loop of intervals $f$-cover

#### Citation

Szuca, Piotr. Loops of intervals and Darboux Baire 1 fixed point problem. Real Anal. Exchange 29 (2003), no. 1, 205--209. https://projecteuclid.org/euclid.rae/1149860185