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).