Forward continuity. (English) Zbl 1230.40001

Çakalli, Hüseyin

J. Comput. Anal. Appl. 13, No. 2, 225-230 (2011).

The author defines a sequence \(u_n \in\mathbb R\) to be “forward convergent” (forwardly …?, forward-convergent?) to \(l\) if \(u_{n+1} - u_n \rightarrow l\). Let \(E\subset\mathbb R\). Accordingly, a real function \(f\) is called “forward continuous” on \(E\) if the sequence \(f(x_n)\) is forward convergent to \(0\) whenever \(x_n \in E\) is forward convergent to \(0\). (This property of \(f\) reduces to ordinary continuity of \(f\) on \(E\) if “forward convergent to \(0\)” is replaced throughout by “convergent”.) \(E\) is defined to be “forward compact” if, in any sequence \(x_n \in E\), there is a subsequence forward convergent to \(0\). (This property of \(E\) reduces to ordinary compactness of \(E\) if “forward convergence to \(0\)” is replaced by “convergent in \(E\)”). The main result reads: If \(f\) is forward continuous on a forward compact set \(E\), then it is uniformly continuous on \(E\). Forward continuity is confronted with continuity that is based on statistical convergence and on the convergence of sequence matrix transforms.

Reviewer: Hans-Heinrich Körle (Marburg)

MSC:

40A05	Convergence and divergence of series and sequences
26A15	Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable

Keywords:

forward compact; forward convergent; forward continuity