I'm learning about functions that satisfy Holder's Condition of order $\alpha$. Specifically,
A function $f$ is said to satisfy a Hölder condition of order $\alpha > 0$ if there exist $M$ such that
$$|f(y) - f(x)| \leq M|x-y|^{\alpha} \qquad \forall x,y \in [a,b]$$
For a little context, I have shown the following two statements:
1) $f$ satisfies a Hölder condition of order $\alpha > 1 \implies f$ is constant on $[a,b]$
2) $f$ satisfies a Hölder condition of order $\alpha = 1 \implies f \in BV([a,b])$
Now, I am trying to find a function that satisfies the a Hölder condition of order $\alpha < 1$ and is not of bounded variation. My possible candidate is $f:[0,1] \to \mathbb{R}$ where $f$ is defined as follows: $$f(x) = \sqrt{x}\sin(\frac{1}{x}).$$ I have showed that $f$ is not of bounded variation on $[0,1]$. However, I am having trouble showing that it satisfies a Hölder condition of order $\alpha < 1$. Any help is greatly appreciated. Thanks!