Construction of functions with given cluster sets

TL;DR

This paper investigates the construction of functions with prescribed cluster sets on the boundary of their domain within topological spaces, extending previous results in the area.

Contribution

It provides new methods to construct functions with specified cluster sets on the boundary of their domain in metrizable topological spaces.

Findings

Existence of functions with prescribed cluster sets on boundary points.

Construction techniques for functions with specific boundary behavior.

Extension of previous results to more general topological spaces.

Abstract

In this paper we continue our research of functions on the boundary of their domain and obtain some results on cluster sets of functions between topological spaces. In particular, we prove that for a metrizable topological space $X$, a dense subspace $Y$ of a metrizable compact space $\overline{Y}$, a closed nowhere dense subset $L$ of $X$, an upper continuous compact-valued multifunction ${\Phi:L\multimap \overline{Y}}$ and a set $D\subseteq X\setminus L$ such that $L\subseteq \overline{D}$, there exists a function $f:D\to Y$ such that the cluster set $\overline{f} (x)$ is equal to $\Phi (x)$ for any $x\in L$.