# lipschitz property of the derivative of a convex function

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).

-
Cross-posted to MSE: math.stackexchange.com/questions/116860/… –  Captain Oates 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