Abstract

For a real valued functions on Euclidean space, the following three conditions are equivalent:

(1) it is Lebesgue measurable,

(2) it is approximately continuous Lebesgue almost everywhere,

(3) outside a set of arbitrarily small Lebesgue measure, it agrees with a continuous function.

We will survey the theory that emerges when replacing the continuity condition with that of being k times continuously differentiable. Convex functions (for k=2) and k times weakly differentiable functions are examples of functions satisfying the resulting conditions.