## Real Analysis Exchange

- Real Anal. Exchange
- Volume 33, Number 1 (2007), 145-152.

### Orbits of Darboux-Like Real Functions

#### Abstract

We show that, with respect to the dynamics of iteration, Darboux-like functions from $\mathbb{R}$ to $\mathbb{R}$ can exhibit some strange properties which are impossible for continuous functions. To be precise, we show that (i) there is an extendable function from $\mathbb{R}$ to $\mathbb{R}$ which is `universal for orbits' in the sense that it possesses every orbit of every function from $\mathbb{R}$ to $\mathbb{R}$ up to an arbitrary small translation, and which has orbits asymptotic to any real sequence, (ii) there is a function $f\:mathbb{R}\to \mathbb{R}$ such that for every $n\in \mathbb{N}$, $f^n$ is almost continuous and the graph of $f^n$ is dense in $\mathbb{R}^2$, in spite of the fact that all $f$-orbits are finite. To prove (i) we assume the Continuum Hypothesis.

#### Article information

**Source**

Real Anal. Exchange, Volume 33, Number 1 (2007), 145-152.

**Dates**

First available in Project Euclid: 28 April 2008

**Permanent link to this document**

https://projecteuclid.org/euclid.rae/1209398384

**Mathematical Reviews number (MathSciNet)**

MR2402869

**Zentralblatt MATH identifier**

1132.54023

**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} 26A18: Iteration [See also 37Bxx, 37Cxx, 37Exx, 39B12, 47H10, 54H25] 54H20: Topological dynamics [See also 28Dxx, 37Bxx]

**Keywords**

Darboux-like function orbit topological transitivity real sequence continuum hypothesis

#### Citation

Moothathu, T. K. Subrahmonian. Orbits of Darboux-Like Real Functions. Real Anal. Exchange 33 (2007), no. 1, 145--152. https://projecteuclid.org/euclid.rae/1209398384