From Uniform Continuity to Absolute Continuity

TL;DR

This paper establishes a sufficient condition under which a uniformly continuous function on an interval is also absolutely continuous, specifically when the function is piecewise convex, bridging a gap between these two concepts.

Contribution

It introduces a new theorem that guarantees absolute continuity for uniformly continuous, piecewise convex functions on real intervals, expanding understanding of function properties.

Findings

Uniform continuity does not imply absolute continuity in general.

Piecewise convexity combined with uniform continuity ensures absolute continuity.

The paper provides a clear sufficient condition linking these continuity concepts.

Abstract

Absolute continuity implies uniform continuity, but generally not vice versa. In this short note, we present one sufficient condition for a uniformly continuous function to be absolutely continuous, which is the following theorem: For a uniformly continuous function f defined on an interval of the real line, if it is piecewise convex, then it is also absolutely continuous.