Characterizations and generalizations of continuity

Abstract:

The condition $f(x + 2h) - 2f(x + h) + f(x) = o(1)$ (as $h \to 0$) at each x is equivalent to continuity for measurable functions. But there is a discontinuous function satisfying $2f(x + 2h) - f(x + h) - f(x) = o(1)$ at each x. The question of which generalized Riemann derivatives of order 0 characterize continuity is studied. In particular, a measurable function satisfying $\sum \nolimits _{i = 1}^n {{\alpha _i}f(x + {\beta _i}h) \equiv 0}$ must be a polynomial. On the other hand, for any Riemann derivative of order 0 and any $p \in [1,\infty ]$, generalized ${L^p}$ continuity is equivalent to ${L^p}$ continuity almost everywhere.