- #1
jamilmalik
- 14
- 0
Hello everyone,
I was wondering if I could get a simple introduction to this Theorem since I will have to be giving a presentation on it within the next month. Based on the statement itself, there is an assumption made in the hypothesis which is something I haven't quite understood yet:
If ##U## is an open subset of ##\mathbb{R^n}## and ##f:U \to \mathbb{R^m}## 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.
What exactly is Lipschitz continuous? I asked a professor of mine and he said to think about it as a bounded slope which makes sense looking at the definition of Lipschitz continuity. However, could someone please provide a thorough explanation of this? For instance, how would one go about proving this Theorem?
I was wondering if I could get a simple introduction to this Theorem since I will have to be giving a presentation on it within the next month. Based on the statement itself, there is an assumption made in the hypothesis which is something I haven't quite understood yet:
If ##U## is an open subset of ##\mathbb{R^n}## and ##f:U \to \mathbb{R^m}## 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.
What exactly is Lipschitz continuous? I asked a professor of mine and he said to think about it as a bounded slope which makes sense looking at the definition of Lipschitz continuity. However, could someone please provide a thorough explanation of this? For instance, how would one go about proving this Theorem?