New Definitions of Continuity

Abstract

We classify all generalized $\mathcal{A}$-differences of any order $n\geq0$ for which $\mathcal{A}$-continuity at $x$ implies ordinary continuity at $x$. We show that the only $\mathcal{A}$-continuities that are equivalent to ordinary continuity at $x$ correspond to the limits of the form \[ \lim_{h\rightarrow0}A\left[ f(x+rh) +f\left( x-rh\right) -2f(x)\right] +B\left[ f\left( x+sh\right) -f\left( x-sh\right) \right] , \] with $ABrs\neq0$. All other $\mathcal{A}$-continuities truly generalize ordinary continuity.