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What type of growth conditions can establish \$||f(x)||\geq K \text{dist}(x,\mathcal{X})\$?


New member
Jun 5, 2018
Hello! I have the following question:

Let $f:\mathbb{R}^n\rightarrow \mathbb{R}^n$. Is there any class of class of functions and some kind of "growth conditions" such that bounds like below can be established:
||f(x)||\geq g\left( \text{dist}(x,\mathcal{X})\right),
with $\mathcal{X}$ := $\{x:f(x)=0\}$ (zero set of $f$) and some function $g$ (e.g. $\ell 2$-norm).

I am interested to know the class of functions. Any help will help a lot. Thanks in advance
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