Let $u:[0,1] \rightarrow \mathbb{R}$ be defined by $$ u(x):=\left\{\begin{array}{ll} x^{a} \sin \frac{1}{x^{b}} & \text { if } 0<x \leq 1 \\ 0 & \text { if } x=0 \end{array}\right. $$ where $a, b \in \mathbb{R}$. Study for which $a, b$ the function $u$ has bounded pointwise variation. Furthermore, when $u$ has bounded pointwise variation, must it be Lipschitz continuous?
Well, it's easy to show that $u(x)$ is continuous for all $a\not =0\in \mathbb{R}$. Actually, I did some analysis for some special $a,b$ and got some result, but I think it's really difficult for me to find the general relation between $a$ and $b$.
Can somebody give me any hint or detailed solution?