New definitions of continuity
In the present work, the authors classify all generalized \(A\)-differences of any order \(n\geq 0\) for which \(A\)-continuity at \(x\in \mathbb{R}\) implies ordinary continuity at \(x\). A function \(f:\mathbb{R}\to \mathbb{R}\) is called \(A\)-continuous at \(x\in \mathbb{R}\) if for \(A=\{{{A}_{1}},\dots,{{A}_{1}},{{a}_{1}},\dots,{{a}_{m}}\}\), \(m\geq 2\), \(\sum\limits_{k=1}^{m}{{{A}_{k}}}=0\), \({{a}_{k}}\neq {{a}_{j}}\), \(\;k\neq j\), \(k,j=1,\ldots,m\) the relation \({{\Delta}_{A}}f(x,h):=\sum\limits_{k=1}^{m}{{{A}_{k}}}f(x+h{{a}_{k}})=o(1)\) as \(h\to 0\) holds. It is proved in the case \(m=2\), that the only \(A\)-continuities that are equivalent to ordinary continuity at \(x\) correspond to the limits of the form \(\underset{h\to 0}{\lim}A[f(x+rh)+f(x-rh)-2f(x)]+B[f(x+sh)-f(x-sh)],\) with \(ABrs\neq 0\). Note that all other \(A\)-continuities truly generalize ordinary continuity.
scientific article; zbMATH DE number 6848844