Rademacher's theorem
In mathematical analysis, Rademacher's theorem, named after Hans Rademacher, states the following: If U is an open subset of Rn and f : U → Rm is Lipschitz continuous, then f is differentiable almost everywhere in U; that is, the points in U at which f is not differentiable form a set of Lebesgue measure zero.
Generalizations
There is a version of Rademacher's theorem that holds for Lipschitz functions from a Euclidean space into an arbitrary metric space in terms of metric differentials instead of the usual derivative.
gollark: I mean, I have code for ARish stuff *anyway*, but if I make one it'll be significantly easier to make an *eeeevil* one.
gollark: Basically, I can't stop the code from being used to make an *un*-time-limited ARer.
gollark: https://forums.dragcave.net/topic/183544-yet-another-hatchery/
gollark: They can probably be viewbombed regardless of well-programmed-ness, unless they have rate limits.
gollark: Apparently quite a few people there ended up just trying to bodge together PHP code from examples they'd seen, which is probably not a good foundation for reliable code and stuff. I assume that most *running* hatcheries are OK, though.
See also
References
- Federer, Herbert (1969), Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, 153, Berlin–Heidelberg–New York: Springer-Verlag, pp. xiv+676, ISBN 978-3-540-60656-7, MR 0257325, Zbl 0176.00801. (Rademacher's theorem is Theorem 3.1.6.)
- Heinonen, Juha (2004). "Lectures on Lipschitz Analysis" (PDF). Lectures at the 14th Jyväskylä Summer School in August 2004. (Rademacher's theorem with a proof is on page 18 and further.)
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.