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1. • CommentRowNumber1.
   • CommentAuthorTobyBartels
   • CommentTimeFeb 8th 2014
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   Author: TobyBartels
   Format: MarkdownItexI was disappointed to discover that [[Boman\'s theorem]] doesn\'t work as one would like for $C^k$ functions with $0 \lt k \lt \infty$. So I wrote up something about it. (This is all in Boman\'s 1967 paper; he covered everything in 20 freely accessible pages!)

   I was disappointed to discover that Boman's theorem doesn't work as one would like for $C^k$ functions with $0 \lt k \lt \infty$. So I wrote up something about it. (This is all in Boman's 1967 paper; he covered everything in 20 freely accessible pages!)

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeFeb 8th 2014
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   Author: Mike Shulman
   Format: MarkdownItexIntriguing!

   Intriguing!

3. • CommentRowNumber3.
   • CommentAuthorAndrew Stacey
   • CommentTimeFeb 8th 2014
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   Author: Andrew Stacey
   Format: MarkdownItexIf I remember right, though, it _does_ work for Lipschitz functions (of arbitrary finite order).

   If I remember right, though, it does work for Lipschitz functions (of arbitrary finite order).

4. • CommentRowNumber4.
   • CommentAuthorTobyBartels
   • CommentTimeFeb 9th 2014
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   Author: TobyBartels
   Format: MarkdownItexRight; I put in a comment that Boman had stronger results involving Lipschitz conditions but didn\'t write them up. Not only does it work as you\'d expect when the differentiability condition is a Lipschitz differentiability condition; but even in the result where the order of differentiability must change, you get a Lipschitz condition in the conclusion for free. (For details, see Theorem 2 in Boman 1967, which is linked from the article.)

   Right; I put in a comment that Boman had stronger results involving Lipschitz conditions but didn't write them up.

   Not only does it work as you'd expect when the differentiability condition is a Lipschitz differentiability condition; but even in the result where the order of differentiability must change, you get a Lipschitz condition in the conclusion for free. (For details, see Theorem 2 in Boman 1967, which is linked from the article.)