  The truncated variation, $TV^c$, is a fairly new concept introduced in [5].
Roughly speaking, given a c\`adl\`ag function $f$, its truncated variation is
"the total variation which does not pay attention to small changes of $f$,
below some threshold $c>0$". The very basic consequence of such approach is
that contrary to the total variation, $TV^c$ is always finite. This is
appealing to the stochastic analysis where so-far large classes of processes,
like semimartingales or diffusions, could not be studied with the total
variation. Recently in [6], another characterization of $TV^c$ was found.
Namely $TV^c$ is the smallest total variation of a function which approximates
$f$ uniformly with accuracy $c/2$. Due to these properties we envisage that
$TV^c$ might be a useful concept to the theory of processes.
  For this reason we determine some properties of $TV^c$ for some well-known
processes. In course of our research we discover intimate connections with
already known concepts of the stochastic processes theory.
  Firstly, for semimartingales we proved that $TV^c$ is of order $c^{-1}$ and
the normalized truncated variation converges almost surely to the quadratic
variation of the semimartingale as $c\searrow0$. Secondly, we studied the rate
of this convergence. As this task was much more demanding we narrowed to the
class of diffusions (with some mild additional assumptions). We obtained the
weak convergence to a so-called Ocone martingale. These results can be viewed
as some kind of large numbers theorem and the corresponding central limit
theorem.
  All the results above were obtained in a functional setting, viz. we worked
with processes describing the growth of the truncated variation in time.
Moreover, in the same respect we also treated two closely related quantities -
the so-called upward truncated variation and downward truncated variation.
