2
$\begingroup$

Is there a variant of Rademacher‘s Theorem where the smallness of the points of non-differentiability is measured in terms of Baire category instead of measure?

More precisely, let X be a separable Banach space and Y be a space with the RNP. Moreover, let $U\subset X$ be an open set and $f\colon U\to Y$ be a Lipschitz mapping. Is the set of points where $f$ is not differentiable a meager set?

As a partial question: Is something known about the above in finite dimensions?

Improve this question
$\endgroup$

1 Answer 1

Reset to default
5
$\begingroup$

No, not even in one dimension. Googling "not differentiable on a residual set" yields a citation of the paper F. Mignot, Contrôle dans les inéquations variationelles elliptiques, J. Functional Analysis 22 (1976), 130–185.

Improve this answer
$\endgroup$
1
  • $\begingroup$ Thanks, the counter example is indeed given in that paper - it can be found in Remarque 3. $\endgroup$
    – Christian
    Commented May 23, 2018 at 18:59

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .