A continuous function on $[0,1]$ not of bounded variation I'm looking for a continuous function $f$ defined on the compact interval  $[0,1]$ which 
is not of bounded variation.
I think such function might exist. Any idea?
Of course the function $f$ such that
$$
f(x) =
\begin{cases}
1  & \text{if $x \in [0,1] \cap \mathbb{Q}$} \\\\
0 & \text{if $x \notin [0,1] \cap \mathbb{Q}$}
\end{cases}
$$
is not of bounded variation on $[0,1]$, but it is not continuous  on $[0,1]$.

For example $x\sin(1/x)$ (and 0 for $x=0$).
The variation is unbounded, because it is bounded ;-) from below by the sum of absolute values of extrema, but it is the harmonic-like series.
