POTW A Function in the Continuous Hölder Class

[Euge]

Let $0 < \alpha < 1$. Find a necessary and sufficient condition for the function $f : [0,1] \to \mathbb{R}$, $f(x) = \sqrt{x}$, to belong to the class $C^{0,\alpha}([0,1])$.

 

[Office_Shredder]

I feel like I must be doing something wrong since this is unanswered and I'm the graduate section.

To be in $C^{0,\alpha}([0,1])$ is equivalent to $|f(x)-f(y)|\leq |x-y|^{\alpha}$ for all $x,y\in[0,1]$.

First we observe if $z$ is positive, then $|\sqrt{x+z}-\sqrt{y+z}| \leq |\sqrt{x}-\sqrt{y}|$ since the square root function is concave (this is just algebraically expressing that line segments have smaller slopes as you move to the right).

Applying this to $0$ and $|y-x|$ we get that $|\sqrt{x}-\sqrt{y}|\leq |\sqrt{|x-y|}-\sqrt{0}|=|x-y|^{1/2}$. Hence f is Holder continuous for every $\alpha \leq 1/2$.

On the other hand, if $\alpha>1/2$, $|f(x)-f(0)|=\sqrt{x}>x^{\alpha}$ since for $x\leq 1$ fixed, $x^{\alpha}$ is a decreasing function of $\alpha$. So the square root function is in $C^{0,\alpha}([0,1])$ if and only if $\alpha \leq 1/2$