Sequential definitions of continuity for real functions. (Q1414969)

A function \(f:\mathbb{R}\to \mathbb{R}\) is continuous at a point \(p\in\mathbb{R}\) if and only if \(\lim_{n\to\infty} x_n= p\) implies that \(\lim_{n\to\infty}f(x_n)= f(p)\). If we replace here the usual convergence by another type of convergence we get ``a new type of continuity''. In 1948, \textit{H. Robbins} and \textit{E. C. Buck} replaced the usual convergence by the Cesàro limitation and obtained so the ``Cesàro continuity'' which gives the linearity of \(f\) [Am. Math. Mon. 53, 470--471 (1946); 55, 36 (1948), Problem 4216]. In the following years several authors continued in this study and obtained new types of continuity that led in most cases to the linearity or usual continuity of \(f\). The authors of this paper unify these types of continuity into the concept of \(G\)-continuity, where \(G\) is a linear functional. For \(G\) they take \(A\)-limitation (\(A\) being a matrix), almost convergence and statistical convergence. The relations of \(G\)-continuities to linearity and usual continuity are investigated in this paper.