To show that a function is not of bounded variation on [0,1] What would be a sequence of partitions of [0,1] to show that the function 

$f(x)$ = \begin{cases}
0,  &amp; \text{if $x$ $\in$[0,1]$\cap$$Q^c$} \\
-x, &amp; \text{if $x$$\in$[0,1]$\cap$$Q$}
\end{cases}

is not of bounded variation on [0,1]?
Try partition

$x_n    = \frac{1}{n}$ , if $n$ is odd

$=\frac{1}{n+n^\frac{1}{n}}$ , if n is even.
Try partition

$x_n    = \frac{1}{n}$ , if $n$ is odd

$=\frac{1}{n+n^\frac{1}{n}}$ , if n is even.