CHARACTERIZATIONS AND GENERALIZATIONS OF CONTINUITY

The condition f(x + 2h) - 2f(x + h) + f(x) = o(1) (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(1 ) at each x. The question of which generalized Riemann derivatives of order 0 characterize continuity is studied. In particular, a measurabl e function satisfying SIGMA(i=1)n alpha(i)f(x + beta(i)h) = 0 must be a polynomial. On the other hand, for any Riemann derivative of order 0 and any p is-an-element-of [1, infinity], generalized L(p) continuity is equivalent to L(p) continuity almost everywhere.