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.

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.

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