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Is there an absolutely continuous function $f$ satisfying
$$ |f(x+\delta)+f(x-\delta)-2f(x)|\leq \mbox{const}\frac{|\delta|}{\log \frac{1}{|\delta|}},\,\,\, |\delta|<1, $$ which is not $C^{1}$?

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  • $\begingroup$ What about the function that maps $x$ to 0 if $x=0$, or else to $\frac{|x|}{\log\frac{1}{|x|}}$? $\endgroup$ Nov 6, 2013 at 8:45
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    $\begingroup$ Dear Ravi, maybe you could add some motivation for this question, together with some general information on related known facts. $\endgroup$ Nov 6, 2013 at 11:16
  • $\begingroup$ This question seems more appropriate for MSE than MO. $\endgroup$
    – user42090
    Nov 6, 2013 at 12:50
  • $\begingroup$ Is it clear that it must be differentiable? $\endgroup$
    – username
    Nov 6, 2013 at 13:14
  • $\begingroup$ @ Delio M. No. $x=\delta=\frac{1}{2}$ $\endgroup$
    – Ravi
    Nov 7, 2013 at 0:47

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