Vol. 111, No. 2, 1984

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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

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
Received: 26 April 1982
Published: 1 April 1984
William H. Julian
Department of Mathematical Sciences
New Mexico State University
Las Cruces NM 88003
United States
Fred Richman