Simultaneous Extension of Continuous and Uniformly Continuous Functions

TL;DR

This paper demonstrates that classical extension theorems for continuous functions also preserve uniform continuity, providing explicit constructions and refining existing results with new extension operators.

Contribution

It shows that various known extension formulas can be adapted to preserve uniform continuity, answering a longstanding question and refining Dugundji's theorem.

Findings

Extension constructions preserve uniform continuity.

Explicit formulas for continuous and uniformly continuous extensions.

Refined extension operators that are nonlinear but preserve uniform continuity.

Abstract

The first known continuous extension result was obtained by Lebesgue in 1907. In 1915, Tietze published his famous extension theorem generalising Lebesgue's result from the plane to general metric spaces. He constructed the extension by an explicit formula involving the distance function on the metric space. Thereafter, several authors contributed other explicit extension formulas. In the present paper, we show that all these extension constructions also preserve uniform continuity, which answers a question posed by St. Watson. In fact, such constructions are simultaneous for special bounded functions. Based on this, we also refine a result of Dugundji by constructing various continuous (nonlinear) extension operators which preserve uniform continuity as well.