Hölder Continuous Maps from ##R## to a Metric Space

In summary, a Hölder continuous map is a type of mathematical function that is characterized by a certain level of smoothness and lack of abrupt changes. It is stronger than other forms of continuity and has important implications in the study of metric spaces, analysis, and various applications in mathematics, physics, and engineering. An example of a Hölder continuous map is the function f(x) = √x, and it is used in the study of dynamical systems, fractals, and partial differential equations.
  • #1
Euge
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Let ##\gamma > 1##. If ##(X,d)## is a metric space and ##f : \mathbb{R} \to X## satisfies ##d(f(x),f(y)) \le |x - y|^\gamma## for all ##x,y\in \mathbb{R}##, show that ##f## must be constant.
 
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  • #2
Hint: If ##a<b## with ##f(a)\neq f(b)##, chop up the interval ##[a,b]## into many small pieces.
 
  • #3
Since this is a POTW, if you have a solution, @Infrared, please don't hesitate to post it! :-)
 
  • #4
Oh I generally don't give solutions here because I'm past the "university student" level,

Without loss of generality, I just check that ##f(0)=f(1)## to make the algebra nicer.
Let ##0=t_0<t_1<\ldots<t_n=1## be the partition ##t_k=\frac{k}{n}.## The given condition is ##d(f(t_i),f(t_{i+1})\leq 1/n^{\gamma}.## Summing over all consecutive ##t_i## and using the triangle inequality gives

$$d(f(0),f(1))\leq\sum_{k=0}^{n-1} d(f(t_k),f(t_{k+1}))\leq \frac{n}{n^{\gamma}}=n^{1-\gamma}.$$

As ##n\to\infty,## the right term goes to 0, so the distance between ##f(0)## and ##f(1)## has to be zero too.
 
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