A uniformly continuous function on [0, 1] that is everywhere different from its infimum

William H. Julian and Fred Richman

Vol. 111 (1984), No. 2, 333–340

DOI: 10.2140/pjm.1984.111.333

Abstract

An example of a uniformly continuous function on [0,1] that is everywhere different from its infimum is constructed in the context of Bishop’s constructive mathematics using a consequence of Chruch’s thesis. The existence of such a function is shown to be equivalent to the constructive denial of König’s lemma. Conversely König’s lemma is shown to be equivalent to the intuitionistic theorem that every positive uniformly continuous function on [0,1] has a positive infimum. Various applications to constructive mathematics are given.

Mathematical Subject Classification 2000

Primary: 03F60

Secondary: 26A15

Milestones

Received: 26 April 1982

Published: 1 April 1984