Constructive Continuity of Increasing Functions

Abstract

Let f be an increasing real-valued function defined on a dense subset D of an interval I. The continuity of f is investigated constructively. In particular, it is shown that for each compact interval $\left[ a,b\right]$ which has end points in D and is contained in the interior of I,  and for each $\varepsilon >0,$ there exist points $x_{1},\ldots ,x_{M}$ of $\left[ a,b\right]$ such that $f(x^{+})-f(x^{-})<\varepsilon$ whenever $x\in \left( a,b\right)$ and $x\ne x_{n}$ for each n. As a consequence, there exists a sequence $(x_{n})_{n\ge 1}$ in I such that f is continuous at each point of D that is distinct from each $x_{n}$.

For Vladik Kreinovich, on the occasion of his 65th birthday.