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. )
Best Answer
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.