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Is there a differentiable (but not continuously differentiable) function f between Euclidean spaces (or manifolds, whatever) such that the set of critical values of f has nonzero measure?

Twigg
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  • Would such a thing even be considered differentiable? You could build a piecewise function that is not differentiable anywhere on some set and then differentiable on the complement of said set – ClassicStyle Apr 21 '15 at 15:03
  • I guess this is a dumb question when looking up Sard's theorem, but why isn't the zero map a counterexample? –  Apr 21 '15 at 15:15
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    @avid19 the set of critical value of the zero function is one point: zero. Critical value is the value of the function at a critical point. – user126154 Apr 21 '15 at 15:19
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    related http://mathoverflow.net/questions/167323/everywhere-differentiable-function-that-is-nowhere-monotonic – user126154 Apr 21 '15 at 15:19
  • @user126154 oh. Thank you. :) –  Apr 21 '15 at 15:24
  • @TylerHG I agree with you that it's pretty hard to imagine such a function being differentiable, but this seems like the kind of thing that could pop up in a book like Counterexamples in Analysis. – Twigg Apr 21 '15 at 15:32
  • Duplicate of https://math.stackexchange.com/q/2391429. There is no such $f$ – Dap Oct 16 '17 at 07:52

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