Continuity of monotone functions
Mark Mandelker · Pacific Journal of Mathematics · 1982
Two refractory problems in modern constructive analysis concern real-valued functions on the closed unit interval: Is every function pointwise continuous?Is every pointwise continuous function uniformly continuous?For monotone functions, some answers are given here.Functions which satisfy a certain strong monotonicity condition, and approximate intermediate values, are pointwise continuous.Any monotone pointwise continuous function is uniformly continuous.Continuous inverse functions are also obtained.The methods used are in accord with the principles of Bishop's Foundations of Constructive Analysis, 1967.