New Definitions of Continuity

We classify all generalized A-differences of any order n ≥ 0 for which A-continuity at x implies ordinary continuity at x. We show that the only A-continuities that are equivalent to ordinary continuity at x correspond to the limits of the form limh →0A[f(x+rh)+f(x-rh)-2f(x)]+B[f(x+sh)-f(x-sh)], with ABrs ≠ 0. All other A-continuities truly generalize ordinary continuity.