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For $X$, $Y$ Banach spaces and an open $U \subseteq X$, say that a function $f : U\to Y$ is:

  • uniformly Lipschitz in $U$ if $$ \sup_{\substack{\mathstrut x,y\in U \\ x\neq y}} \frac{\lVert f(y)-f(x)\rVert}{\lVert y-x\rVert} <\infty\,, $$
  • P-Lipschitz at $a\in U$ if $$ \limsup_{\substack{\mathstrut x,y\to a \\ x\neq y}} \frac{\lVert f(y)-f(x)\rVert}{\lVert y-x\rVert} <\infty\,\text{, and} $$
  • F-Lipschitz at $a\in U$ if $$ \limsup_{x\to a} \frac{\lVert f(x)-f(a)\rVert}{\lVert x-a \rVert} < \infty\,. $$

The relationships among these seem to be like this: $f$ is

  1. Uniformly continuously differentiable in $U$ only if uniformly Lipschitz in $U$;
  2. Peano (= strictly, strongly) differentiable at $a$ only if P-Lipschitz at $a$;
  3. Fréchet differentiable at $a$ only if F-Lipschitz at $a$;
  4. P-Lipschitz at $a$ only if F-Lipschitz at $a$;
  5. F-Lipschitz at $a$ only if continuous at $a$;
  6. uniformly Lipschitz in $U$ only if P-Lipschitz at every point of $U$;
  7. P-Lipschitz at $a$ only if (and thus iff) uniformly Lipschitz in a neighbourhood of $a$;
  8. F-Lipschitz at every point of $U$ iff uniformly Lipschitz on every compact $K\subset U$;
  9. F-Lipschitz at every point of $U$ only if (and thus iff) P-Lipschitz at every point of $U$, provided $X$ is finite-dimensional, but not otherwise.

Are there conventional names, or references, for these three notions of Lipschitzness? Uniform Lipschitz is usually just Lipschitz, but locally Lipschitz seems to be used to refer both to F- and P-Lipschitz, probably because of the last point above in the finite-dimensional setting. Or am I missing some easy equivalence here?

Toby Bartels
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Alex Shpilkin
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  • Locally Lipschitz means that for any point there exists an open neighbourhood so that the function is Lipschitz (what you call uniformly Lipschitz) on this neighbourhood. This is the same as $P$ Lipschitz at every point. Also note that your statement $6$ is false, for example $U=\Bbb R$ and $f:x\mapsto x^2$ is not Lipschitz but locally Lipschitz. – s.harp May 27 '17 at 12:45
  • @s.harp So you mean P-Lipschitz by locally Lipschitz. See the link in the question or this handout for an opposite convention. In any case the problem is exactly that I’m unable to find a worthy source mentioning all three notions (and they kind of make sense, as seen from points 1—3; here are some more intermediate properties). – Alex Shpilkin May 27 '17 at 17:31
  • @s.harp WRT point 6, “only if” is “⇒”, i.e. it’s “Lipschitz ⇒ locally Lipschitz everywhere” in your terms, which is true. – Alex Shpilkin May 27 '17 at 17:32
  • This paper uses F-Lipschitz functions, calling them pointwise Lipschitz. – Alex Shpilkin May 28 '17 at 01:38
  • I see, it seems I somehow misread your point 6 to be an iff statement. Anyway, locally $X$ always means for any point there is a neighbourhood on which $X$. For example a space is locally compact if every point has a compact neighbourhood, a function is locally convex if around any point you have a neighbourhood on which the function is convex, and so on. I don't know any special literature on Lipschitz continuity and its different flavours unfortunately. – s.harp May 28 '17 at 13:09
  • Also, sneakily, Abraham, Marsden and Ratiu (1988) call F-Lischitzness the Lispchitz property in their proposition 2.4.1, which is a (needlessly?) restricted version of my point 3. This term is not used anywhere else in the book as far as I can tell. – Alex Shpilkin May 29 '17 at 18:28
  • For point $3$, what about a function that is $0$ on rationals and identity on irrationals, this is F-Lipschitz at $0$ but not differentiable. – s.harp May 29 '17 at 19:03
  • @s.harp “Only if” again :) – Alex Shpilkin May 29 '17 at 19:39
  • Oh jeeze, reading comprehension fail – s.harp May 29 '17 at 21:22

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