Characterizations and Generalizations of Continuity

The condition f(x + 2h) 2f(x + h) + f(x) = o(l) (as h -* 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(l) at each x. The question of which generalized Riemann derivatives of order 0 characterize continuity is studied. In particular, a measurable function satisfying EZI1if(x + fih) _ 0 must be a polynomial. On the other hand, for any Riemann derivative of order 0 and any p E [1, oo], generalized LP continuity is equivalent to LP continuity almost everywhere.