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?


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.

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

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