Vol. 72, No. 1, 1977

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Approximately differentiable functions: the r topology

Richard John O’Malley

Vol. 72 (1977), No. 1, 207–222

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
Received: 15 October 1976
Revised: 21 March 1977
Published: 1 September 1977
Richard John O’Malley