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##