Approximately differentiable functions: the r topology

Richard John O’Malley

Vol. 72 (1977), No. 1, 207–222

DOI: 10.2140/pjm.1977.72.207

Abstract

It is shown that the coarsest topology making all approximately differentiable functions continuous is not the density topology. The correct topology, the r topology, is introduced, and the structure of the open sets in this topology is examined. Among other things, it is proven that any r-open set must have nonempty Euclidean interior.

In the development of the r topology, two new classes of functions play a role. These classes are the Baire ^∗ 1 approximately continuous functions and the ambivalent approximately continuous functions. For either class, r is also the coarsest topology for which they are continuous.

Mathematical Subject Classification 2000

Primary: 26A24

Milestones

Received: 15 October 1976

Revised: 21 March 1977

Published: 1 September 1977