Forward continuity. (English) Zbl 1230.40001

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.


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