What type of function satisfy a type of growth condition?

In summary, the conversation discusses the possibility of establishing bounds for functions ##f:\mathbb{R}^n\rightarrow\mathbb{R}^n## with certain "growth conditions" and a zero set ##\mathcal{X}##. The speaker is interested in finding a more general class of functions and a established theory on the conditions and properties of such functions, similar to Holder conditions but on the inverse of ##f##.
  • #1
Vulture1991
7
0
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
 
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  • #2
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##.
 
  • #3
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##.
 

1. What is a growth condition in a function?

A growth condition in a function refers to a specific property or behavior that the function exhibits as its input values increase. This can include behaviors such as exponential growth, linear growth, or logarithmic growth.

2. What types of functions satisfy a growth condition?

There are many types of functions that can satisfy a growth condition, depending on the specific condition being considered. Some common examples include polynomial functions, exponential functions, and logarithmic functions.

3. How do I determine if a function satisfies a growth condition?

The best way to determine if a function satisfies a growth condition is to graph the function and observe its behavior as the input values increase. If the graph shows a consistent trend or pattern, then the function likely satisfies a growth condition.

4. Can a function satisfy multiple growth conditions?

Yes, it is possible for a function to satisfy multiple growth conditions. For example, a function could exhibit both linear and exponential growth, depending on the range of input values being considered.

5. Why is it important to understand growth conditions in functions?

Understanding growth conditions in functions can help us make predictions and draw conclusions about the behavior of a system or phenomenon. It can also help us identify and analyze patterns in data, and make informed decisions based on those patterns.

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