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

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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.

 

[Infrared]

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

 

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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.