In my work I came up with a continuous function $f(x)$ on $[0, 1]$ with the following properties:
- $f(0) = 0$,
- $f\left(\frac{1}{n}\right) = 0$ for all natural $n$,
- $\displaystyle{f\left(\frac{1}{2}\left(\frac{1}{n^a} + \frac{1}{(n + 1)^a}\right)\right) = \frac{1}{n^a} }$, where $a$ is some positive real parameter, and
- in all other intervals the function $f(x)$ is linear.
Given that, for what values of parameter $a$ is the function $f(x)$ of bounded variation on $[0, 1]$?