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Let $f\in C^1(\mathbb R^n\to \mathbb R)$ be a convex function. Suppose the equation

$$f(x+\Delta x)-f(x)-(f'(x),\Delta x) \leq A|\Delta x|^2$$

holds for some constant $A>0$, any $x\in \mathbb R^n$ and any sufficiently small $\Delta x\in \mathbb R^n$. Is it true that the derivative $f'(x)$ must be a (locally) lipschitz function?

Only case $n>1$ is interesting for me (case $n=1$ can be easily verified).

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Cross-posted to MSE:… – Yemon Choi Mar 6 '12 at 6:51
In case $f$ is $C^2$, then the restriction to convexity can be dropped, and in fact, you can show an iff. – Suvrit Mar 6 '12 at 7:09
Please don't crosspost between here and MSE, without an explanation. – Scott Morrison Mar 6 '12 at 7:25

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