Rademacher's theorem, named after Hans Rademacher, states that if U is an open subset of real n-dimensional space and f, mapping U into real m-dimensional space, is Lipschitz continuous, then f is differentiable almost everywhere in U, meaning the set of points where f fails to be differentiable has Lebesgue measure zero. Differentiability here means approximability by a linear map to first order, which in particular implies that all of f's coordinate-wise partial derivatives exist at almost every point. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
Classification
Statement FormCharacterization Theorem 1 Connections
Has Statement Form
In Branch
Source Rademacher's Theorem (Wikipedia)
Proved By
Source Rademacher's Theorem (Wikipedia)
Sources
1. Rademacher's Theorem (Wikipedia)
Lead section
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.
In Branch: Analysis, Lead sentence
In mathematical analysis, Rademacher's theorem, named after Hans Rademacher, states the following: If U is an open subset of Rn an
Proved By: Hans Rademacher, Lead
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
View the SourceReader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.