×

On the almost continuity of the diagonal of functions. (English) Zbl 0943.26008

Let \(\mathcal A_0\) be the set of almost continuous mappings of the real line, vanishing at the points of discontinuity. For \(f\in \mathcal A_0\), denote by \(\Delta _{\mathcal A_0}(f)\) the set of \(g\in \mathcal A_0\) such that the diagonal map \(f\Delta g\) defined by \(f\Delta g(x)=(f(x),g(x))\) is almost continuous and vanishes at the points of discontinuity.
For \(f\in \mathcal A_0\), the authors give a characterization of \(\Delta _{\mathcal A_0}(f)\), involving the notion of ac-homotopy (cf. R. J. Pawlak [Real Anal. Exch. 20, No. 2, 805-814 (1995; Zbl 0835.26004)]). Another result states that if \(f,g\in \mathcal A_0\) and \(f\Delta g\) is almost continuous and vanishes at the points of discontinuity, then \(f+g\), \(f\cdot g\), \(\min (f,g)\), and \(\max (f,g)\) are in \(\mathcal A_0\). The paper contains also some other interesting results.

MSC:

26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
54C08 Weak and generalized continuity

Citations:

Zbl 0835.26004
PDFBibTeX XMLCite