On approximation by functions having strong entropy point

Abstract

The paper deals with approximation of functions from the unit interval into itself by means of functions having strong entropy point. For this purpose we define a family of functions having the fixed point property: $Conn_C$ (which is a subfamily of the class $Conn$ introduced in [16]). The main result of the paper is a theorem saying that for any function $f\in Conn_C$ and any point $x_0 \in Fix(f)$ there exist
a ring $R \subset Conn_C$ containing function $ f$ and in the intersection of any "graph
neighbourhood of f" and "neighbourhood of f in topology of uniform convergence"
one can find functions $\xi, \psi  \in R$ having strong entropy point $y_0$ located close to the
point $x_0$ and being a discontinuity point of the function $\xi$ and a continuity point of
the function \psi.