Use this tag for questions related to absolute continuity, which is a smoothness property of functions stronger than that of continuity and uniform continuity.
Absolute continuity is a smoothness property of functions stronger than that of continuity and uniform-continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central operations of calculus—differentiation and integration—expressed by the fundamental theorem of calculus in the framework of riemann-integration. Such generalizations are often formulated in terms of the lebesgue-integral. For real-valued functions on the real line, two interrelated notions appear: absolute continuity of functions and absolute continuity of measures. Those two notions are generalized in different directions. The usual derivative of a function is related to the radon-nikodym derivative, or density, of a measure.
We have the following chains of inclusions for functions over a compact subset of the real line:
absolutely continuous ⊆ uniformly continuous = continuous,
and
continuously differentiable ⊆ Lipschitz continuous ⊆ absolutely continuous ⊆ bounded variation ⊆ differentiable almost everywhere.
