Does limit of "approximate zero set" converge to the zero set?

Jun 2018
3
0
lafayette, IN
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.
 
Last edited:

SlipEternal

MHF Helper
Nov 2010
3,728
1,571
If $f$ is a continuous bijection, this follows trivially. You can probably relax the condition to monotone and continuous and still arrive at the conclusion.
 
Jun 2018
3
0
lafayette, IN
Thank you very much for your answer.
Actually \(\displaystyle f\) here is the gradient of a non-convex function \(\displaystyle g\), i.e. \(\displaystyle f=\nabla g\) which is not monotone, and the zero set is the set of critical points. However, I assume \(\displaystyle g\) is \(\displaystyle \mathcal{C}^\infty\).
Do you have any thought how to approach this?
 
Last edited:
Jun 2018
3
0
lafayette, IN
If $f$ is a continuous bijection, this follows trivially. You can probably relax the condition to monotone and continuous and still arrive at the conclusion.
Thank you very much for your answer.
Actually \(\displaystyle f\) here is the gradient of a non-convex function \(\displaystyle g\), i.e. \(\displaystyle f=\nabla g\) which is not monotone, and the zero set is the set of critical points. However, I assume \(\displaystyle g\) is \(\displaystyle \mathcal{C}^\infty\).
Do you have any thought how to approach this?