It is well known that e.g. $sin(1/x)$ is of [unbounded total variation](http://en.wikipedia.org/wiki/Bounded_variation#Examples) (in the interval [0,1] assuming $f(0)=0$). (Preliminary numerical tests suggest that) it is also of unbounded [quadratic variation](http://en.wikipedia.org/wiki/Quadratic_variation). $x\ sin(1/x)$ is of unbounded total variation too but (preliminary numerical tests suggest that) it is of zero quadratic variation. 


**My question:** How to construct a deterministic function with unbounded total variation and bounded (non zero) quadratic variation along these lines? I don't want to have a function which is defined by a sum of terms (like the [Weierstrass function](http://en.wikipedia.org/wiki/Weierstrass_function)) but one which is defined straight forward like the two above mentioned examples. References (if available) would also be appreciated!


**Addendum:** If some of these conjectures are not true please tell me. And please tell me also if it is not possible to construct such a deterministic function (and why not).



Example for deterministic function with unbounded total variation and bounded quadratic variation
Function $x^a\sin(1/x)$ on $(0,1]$ has bounded variation iff $a>1$ and finite quadratic variation iff $a>1/2$. So for your example, take $1/2 < a \le 1$. It looks like more than numerical tests may be needed to decide questions like this... 


I took "quadratic variation" to mean
$$
\sup \sum\_{j=1}^n |f(x\_j)-f(x\_{j-1})|^2
$$
with sup over all finite sequences $x\_0\lt x\_1\lt\cdots\lt x\_n$ in $(0,1]$. Perhaps you meant something else? 


**addition**  

On the other hand, maybe you mean
$$
\lim\_{\delta \to 0}\sup \sum\_{j=1}^n |f(x\_j)-f(x\_{j-1})|^2
$$
with sup over all finite sequences $x\_0\lt x\_1\lt\cdots\lt x\_n$ in $(0,1]$ such that $x\_j-x\_{j-1}<\delta$. In that case, on any inverval where $f$ is monotone it has quadratic variation $0$, so you won't find any such simple example.


