  For a real cadlag function $f$ and positive constant $c$ we find another
cadlag function, which has the smallest total variation possible among the
functions uniformly approximating f with accuracy c=2. The solution is
expressed with the truncated variation, upward truncated variation and downward
truncated variation introduced in the papers R. \L ochowski, Truncated
variation of Brownian motion with drift, Bull. Pol. Acad. Sci. Math. Vol. 56
(2008) and R. \L ochowski, Truncated variation, upward truncated variation and
downward truncated variation of Brownian motion with drift - their
characteristics and applications Stochastic Processes and their Applications
121 (2011). They are analogs of Hahn-Jordan decomposition of a cadlag function
with finite total variation but are always finite even if the total variation
is infinite. We apply obtained results to general stochastic processes with
c?adl?ag trajectories and in the special case of Brownian motion with drift we
apply them to obtain full characterisation of its truncated variation by
calculating its Laplace transform. We also calculate covariance of upward and
downward truncated variations of Brownian motion with drift. Keywords: total
variation, truncated variation, uniform approximation, Brow- nian motion,
Laplace transform
