Let $\displaystyle f:\mathbb{R}^m\rightarrow\mathbb{R}^m$.

Define the zero set by $\displaystyle \mathcal{Z}\triangleq\{x\in\mathbb{R}^m | f(x)=\mathbf{0}\}$ and an $\displaystyle \epsilon$-approximation of this set by $\displaystyle \mathcal{Z}_\epsilon\triangleq\{x\in\mathbb{R}^m|~ ||f(x)||\leq\epsilon\}$ for some $\displaystyle \epsilon>0$. Clearly $\displaystyle \mathcal{Z}\subseteq \mathcal{Z}_\epsilon$. Can one assume any condition on the function $\displaystyle f$ so that

$\displaystyle

\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):

$\displaystyle

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.