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Let $ f: \mathbf{R}^{m} \rightarrow \mathbf{R} $ be a continuous function, $ \omega $ be a modulus of continuity and assume $$ | f(x+h) +f(x-h) -2f(x) | \leq \omega(|h|)|h| $$ whenever $ x,h \in \mathbf{R}^{m} $ and $ 0 \leq |h | \leq 1 $.

Is it known some sharp condition on $ \omega $ in order to have that $ f $ is continously differentiable?

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  • $\begingroup$ Is $\omega$ a modulus of continuity of function $f$ itself? This looks somehow strange. $\endgroup$
    – Fedor Petrov
    Commented Mar 14, 2016 at 9:08
  • $\begingroup$ No, it is just a modulus of continuity for this second order difference quotient. If $ \omega $ is a generic modulus of continuity you get the class of smooth functions in the sense of Zygmund, that are known to be non differentiable a.e. in general. $\endgroup$
    – Longyearbyen
    Commented Mar 14, 2016 at 9:13
  • $\begingroup$ If we do not require additionally that $f$ is continuous or something like that, it may appear that $f$ is additive discontinuous function. $\endgroup$
    – Fedor Petrov
    Commented Mar 14, 2016 at 9:22
  • $\begingroup$ you are right. f has to be assumed to be continuous. I added it. $\endgroup$
    – Longyearbyen
    Commented Mar 14, 2016 at 9:28
  • $\begingroup$ related question mathoverflow.net/q/143659/6451 $\endgroup$
    – BS.
    Commented Mar 14, 2016 at 11:28

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