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?



Rademacher‘s Theorem and Baire category
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.


