Do two close functions share some local minima?

Question

Let $f,g:\mathbb{R}^n \to \mathbb{R}$ be two differentiable functions. Assume that $\| f -g \|_{\infty} \leq \epsilon$. On what conditions on $f$, for every local minima $x$ of $g$, there is a local minimum $x'$ of $f$, such that, $\| x - x'\| \leq \mathcal{O}(h(\epsilon))$ for some positive function $h(\epsilon) \to 0$ as $\epsilon \to 0$ (or at least $\| x - x'\| \leq \text{const}$)? I thought maybe $\| \nabla f(x) - \nabla g(x)\|_{\infty}\leq \epsilon$ might be sufficient.