Search

In this paper we investigate for which closed subsets P of the real line R there exists a continuous map from P onto P² and, if such a function exists, how smooth can it be. We show that there exists an infinitely many times differentiable function f:R → R² which maps an unbounded perfect set P onto P². At the same time, no continuously differentiable function f:R → R² can map a compact perfect set onto its square. Finally, we show that a disconnected compact perfect set P admits a continuous function from P onto P² if, and only if, P has uncountably many connected components. [ABSTRACT FROM AUTHOR]

The purpose of this article is to give a solution to a problem raised by A. Maliszewski in [5] by showing that any lower semicontinuous function can be represented as a sum of two lower semicontinuous strong Šwiątkowski functions. [ABSTRACT FROM AUTHOR]