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 ofConstructive Analysis, 1967.
