Stability of typical continuous functions with respect to some properties of their iterates

Abstract:

Let $I$ be a real compact interval, and let $C$ be the space of continuous functions $I \to I$ with the uniform metric. For $f \in C$ denote $\nu (f) = {\sup _{x \in I}}(\lim {\sup _{n \to x}}{f^n}(x) - \lim {\inf _{n \to x}}{f^n}(x))$, where ${f^n}$ is the $n$th iterate of $f$. Then for each positive $d$ there is an open set ${C^*}$ dense in $C$ such that the oscillation of $v$ at each point of ${C^*}$ is less than $d$. Consequently, $\nu$ is continuous in $C$ except of the points of a first Baire category set.