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

  1. #1
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    Does limit of "approximate zero set" converge to the zero set?

    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 by Vulture; Jun 5th 2018 at 11:35 AM.
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  2. #2
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    Re: Does limit of "approximate zero set" converge to the zero set?

    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.
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  3. #3
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    Re: Does limit of "approximate zero set" converge to the zero set?

    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 by Vulture; Jun 5th 2018 at 12:27 PM.
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  4. #4
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    Re: Does limit of "approximate zero set" converge to the zero set?

    Quote Originally Posted by SlipEternal View Post
    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?
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