I'm trying with this.
For $\alpha, \beta \in \mathbb{R}_+$ let $f_{\alpha, \beta}: [0,1]$ defined as follow
$$f(x)_{\alpha,\beta}=\left\{\begin{array}{ll}x^{\alpha}\sin(1/x^\beta) &\mbox{if }0<x\leq1 ,\\0&\mbox{if }x=0.\end{array}\right.$$
I want to characterize $\alpha, \beta$ susch that:
1.) $f'$ is continuous at $x=0$
2.) $f$ is of bounder variation in $[0,1]$
For the part 2.), I got that
$$f'(x)=-\beta x^{\alpha-\beta-1} \cdot \cos\left(\frac{1}{x^\beta} \right)+ \alpha x^{\alpha-1} \cdot \sin \left( \frac{1}{x^\beta} \right) $$
if $\alpha> \beta$ then $f'$ is integrable
$$ \int_0^1 |f'| = \int_0^1 \left|-\beta x^{\alpha-\beta-1} \cdot \cos\left(\frac{1}{x^\beta} \right)+ \alpha x^{\alpha-1} \cdot \sin \left( \frac{1}{x^\beta} \right)\right| dx$$ $$ \leq \int_0^1 (\beta x^{\alpha - \beta -1} +\alpha x^{\alpha-1} )dx = \beta\int_0^1 x^{\alpha- \beta -1}dx+ \alpha\int_0^1 x^{\alpha -1}dx= \frac{\beta}{\alpha - \beta} + \frac{\alpha}{\alpha}< \infty$$
At this point I don't know how to continue, I think I have to use the FTC and do some partitions to use the definition of $BV$.
For the point 1.) I'm pretty lose.