IJPAM: Volume 32, No. 1 (2006)

A NOTE ON ITERATIONS OF DARBOUX FUNCTIONS

Kandasamy Muthuvel
Department of Mathematics
University of Wisconsin-Oshkosh
800 Algoma Blvd., Oshkosh, Wisconsin, 54901-8601, USA
e-mail: muthuvel@uwsoh.edu


Abstract.We recently proved in a paper that if $g$ is a continuous function that is nonconstant on every nonempty open interval, and $f$ is a Darboux function such that, for every real number $x,$ $f^{n}(x)=g(x)$ for some positive integer $n$ and the set of all such $n$ is bounded, then $f$ is continuous. In this paper, we give an example to show that the above conclusion is not true if the condition ``the set of all such $n$ is bounded" is dropped. However, it is known that if $g$ is the identity function, then $f$ is continuous and, either $f$ is the identity function or $f=f^{-1}$.

Received: August 14, 2006

AMS Subject Classification: 26A15, 54C30

Key Words and Phrases: Darboux functions, continuous functions

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2006
Volume: 32
Issue: 1