Absolute Continuity

An Absolute Continuity is a mathematical property of function or variable being absolutely continuous.

• • …
• Example(s)
  • Lipschitz Continuity.
• Counter-Example(s)
  • Continuity.
  • Uniform Continuity.

See: Singular Function, Continuous Function.

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References

2017

• (Wikipedia, 2017) ⇒ http://en.wikipedia.org/wiki/Absolute_continuity
  • In calculus, absolute continuity is a smoothness property of functions that is stronger than 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 Lebesgue integration. For real-valued functions on the real line two interrelated notions appear: absolute continuity of functions and absolute continuity of measures. These 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

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