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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9
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$\begingroup$ What about the function that maps $x$ to 0 if $x=0$, or else to $\frac{|x|}{\log\frac{1}{|x|}}$? $\endgroup$– Delio MugnoloNov 6, 2013 at 8:45
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5$\begingroup$ Dear Ravi, maybe you could add some motivation for this question, together with some general information on related known facts. $\endgroup$– Pietro MajerNov 6, 2013 at 11:16
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$\begingroup$ This question seems more appropriate for MSE than MO. $\endgroup$– user42090Nov 6, 2013 at 12:50
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$\begingroup$ Is it clear that it must be differentiable? $\endgroup$– usernameNov 6, 2013 at 13:14
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$\begingroup$ @ Delio M. No. $x=\delta=\frac{1}{2}$ $\endgroup$– RaviNov 7, 2013 at 0:47
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