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

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 Knig's lemma. Conversely Knig'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.