Let $f:\mathbb{R}^m\rightarrow\mathbb{R}^m$. Define the zero set by $\mathcal{Z}\triangleq\{x\in\mathbb{R}^m|f(x)=\mathbf{0}\}$ and an $\epsilon$-approximation of this set by $\mathcal{Z}_\epsilon\triangleq\{x\in\mathbb{R}^m|~||f(x)||\leq\epsilon\}$ for some $\epsilon>0$. Clearly $\mathcal{Z}\subseteq \mathcal{Z}_\epsilon$. Can one assume any condition on the function $f$ so that $$ \lim_{\epsilon\rightarrow 0}~\max_{x\in\mathcal{Z}_\epsilon}~\text{dist}(x,\mathcal{Z})=0, $$ holds?
I know in general this doesn't hold by this example (function of a scalar variable): $$ f(x)=\left\{\begin{align} 0,\quad{x\leq 0}; \\ 1/x,\quad x>0. \end{align} \right. $$
I really appreciate any help or hint. Thank you.
If $f$ is continous and stays away from $0$ outside some fixed compact set then your claim holds.