The condition f ( x + 2 h ) − 2 f ( x + h ) + f ( x ) = o ( 1 ) f(x + 2h) - 2f(x + h) + f(x) = o(1) (as h → 0 h \to 0 ) at each x is equivalent to continuity for measurable functions. But there is a discontinuous function satisfying 2 f ( x + 2 h ) − f ( x + h ) − f ( x ) = o ( 1 ) 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 ∑ i = 1 n α i f ( x + β i h ) ≡ 0 \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 ∈ [ 1 , ∞ ] p \in [1,\infty ] , generalized L p {L^p} continuity is equivalent to L p {L^p} continuity almost everywhere.