On the maximum of two unilaterally continuous regulated functions. (English) Zbl 1094.26003

Grande, Marcin

Real Anal. Exch. 29(2003-2004), No. 2, 781-788 (2004).

Summary: We prove that if \(f\) is the maximum of two unilaterally continuous regulated functions, then the set \[ D_{\text{un}}(f) =\left\{x: f \text{ is not unilaterally continuous at } x\right\} \] is unilaterally isolated and for \(x\in D_{\text{un}}(f)\) the inequality \[ f(x) <\max \left\{f(x+), f(x-)\right\} \] holds. Moreover, for a regulated function \(f\) such that \(D_{\text{un}}(f)\) is isolated and for \(x\in D_{\text{un}}(f)\) the inequality \(f(x) <\max \{f(x+), f(x-)\}\) holds, there are two unilaterally continuous regulated functions \(g,h\) with \(f=\max \{g,h\}\).

MSC:

26A15

Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable

Keywords:

regulated function; jump function; unilateral continuity