# Uniform convergence of difference quotients of a convex function

Let $f(\cdot):\mathbb{R}^n\rightarrow \mathbb{R}$ be a convex function. At $x$ denote the subdifferential by $\partial f(x)$ which is compact and closed. Now, define the approximation of $f$ around a point $x_0$ as follows, $$\hat{f}(x)=f(x_0)+\sup_{g\in \partial f(x_0)} <x-x_0,g>$$

For all $x$, $\hat{f}(x)\leq f(x)$. Also, it is known that, for any $v\in\mathbb{R}^n$, the following holds, $$\lim_{t\rightarrow 0^+} \frac{f(x_0+tv)-\hat{f}(x_0+tv)}{t}=0$$ which is why $\hat{f}$ is an approximation. I am wondering whether the above limit is true uniformly over a bounded set of $v$'s. In particular, is the following true?

Claim: Let $S^{n-1}$ be the unit sphere in $\mathbb{R}^n$. Then, given $\epsilon>0$, there exists $t_0>0$ such that, for all $v\in S^{n-1}$ and for all $t\leq t_0$, one has, $$\frac{f(x_0+tv)-\hat{f}(x_0+tv)}{t}<\epsilon$$

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