Graduate What type of function satisfy a type of growth condition?

[Vulture1991]

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$. Is there any class of function and some type of "growth conditions" such that bounds like below can be established:
\begin{equation}
||f(x)||\geq g\left( \text{dist}(x,\mathcal{X})\right),
\end{equation}
with $\mathcal{X}:= \{x:f(x)=0\}$ (zero set) and some function $g$ (like a homogeneous function).

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

 

[mathman]

For $n=1,$ any of the form $x^p$ where $p\ge 1$ will do. $g(u)=u$ works. This can easily be generalized for $n\gt 1$.

 

[Vulture1991]

Yes, that is true. But I am looking for more general class of function and a well-established theory on the conditions and properties of such functions. This is a bit similar to holder conditions but on the inverse of $f$.