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

[Euge]

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.

 

[Infrared]

Hint: If ##a<b## with ##f(a)\neq f(b)##, chop up the interval ##[a,b]## into many small pieces.

 

[Euge]

Since this is a POTW, if you have a solution, @Infrared, please don't hesitate to post it! :-)

 

[Infrared]

Oh I generally don't give solutions here because I'm past the "university student" level,

Spoiler

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.