It is well known that e.g. $sin(1/x)$ is of unbounded total variation (in the interval [0,1] assuming $f(0)=0$). (Preliminary numerical tests suggest that) it is also of unbounded 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) 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).