Real functions Monotone functions, bounded variation, absolute continuity Concept
Absolute continuity
Summary
A real function on a closed interval is absolutely continuous when small total length of input intervals forces small total change of the output. It is stronger than uniform continuity and stronger than bounded variation. The absolutely continuous functions are exactly the indefinite Lebesgue integrals. So they are the functions for which the fundamental theorem of calculus holds in Lebesgue's form. The analogous notion for measures is the absolutely continuous measure.
Where this leads
26 pages build on this one.
- Differential inclusion
- Arnold tongue
- Lebesgue's theorem on differentiability of monotone functions
- Stein's unbiased risk estimate
- Stein's lemma
- Pontryagin's maximum principle
- Stein's example
- Admissible control
- Khinchin integral
- Bolza problem
- Optimal control
- Goddard problem
- Zermelo's navigation problem
- Singular control
- Trajectory optimization
- Cantor function
- Indirect method by shooting
- Fuller's phenomenon
- Fundamental theorem of calculus for Lebesgue integrals
- Control system
- Relaxed control
- Costate equation
- Sub-Riemannian geodesic problem
- Filippov's existence theorem
- Time-optimal control
- Banach–Zarecki theorem
Further reading
About this page
| Tag | ZT6F4C2YHK |
|---|---|
| URI | m2://ZT6F4C2YHK |
| Kind | concept |
| Writing | W0 stub |
| MSC | 26A46 |
| Same as |
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Used by
- Differential inclusion
- Stein's lemma
- Pontryagin's maximum principle
- Admissible control
- Khinchin integral
- Cantor function
- Fundamental theorem of calculus for Lebesgue integrals
- Control system
- Costate equation
- Sub-Riemannian geodesic problem
- Banach–Zarecki theorem