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The common example for differentiable but not continuously differentiable is $x^2\sin \frac{1}{x}$. This still looks almost everywhere continuously differentiable to my not very trained eyes. Are there examples of almost everywhere differentiable but not almost everywhere continuously differentiable? If so, are there any straightforward conditions (possibly to do with one-sided derivatives) that can be combined with almost everywhere differentiable to give almost everywhere continuously differentiable?

(I am trying to show that the Lipschitz continuous function I am working with is almost everywhere continuously differentiable. Rademacher's theorem says that a Lipschitz continuous function is almost everywhere differentiable. )

Jimmy
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  • Your example should read $x^2 \sin \frac{1}{x}$. When you say almost everywhere, you mean in what sense? Lebesgue measure? – H. H. Rugh Aug 06 '16 at 10:58
  • Thanks for spotting the typo; I've fixed it. Yes I'm referring to everywhere except a set with Lebesque measure zero. – Jimmy Aug 06 '16 at 11:00
  • For the Lipschitz question, not immediately obvious (at least to me). – H. H. Rugh Aug 06 '16 at 11:08

1 Answers1

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Construct a measurable set $A$ with the property that for any non-empty open interval $(a,b)$: $$ 0 < m((a,b)\cap A) < m((a,b)) .$$

See e.g. this construction: Construction of a Borel set with positive but not full measure in each interval

Let $f(x)=m ((0,x)\cap A) - m ((x,0) \cap A)$. Then $f$ is Lipschitz and has an a.e. derivative $f'=1_A(x)$. But in any interval $(a,b)$, $f'$ takes both values 0 and 1 with positive probability. So is nowhere continuous.

Community
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H. H. Rugh
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  • Thanks H. Apologies in advance for the vague question, but can you see any simple conditions that could be combined almost everywhere differentiable to ensure almost everywhere continuously differentiable? – Jimmy Aug 07 '16 at 07:22
  • Not really. I think that I would need more information about the Lipschitz fct you work with to come up with an idea. – H. H. Rugh Aug 07 '16 at 17:29