# almost everywhere differentiable but not almost everywhere continuously differentiable

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. )

• 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 6 '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 6 '16 at 11:00
• For the Lipschitz question, not immediately obvious (at least to me). – H. H. Rugh Aug 6 '16 at 11:08

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