  For a real c\`{a}dl\`{a}g function f and a positive constant c we find
another c\`{a}dl\`{a}g function, which has the smallest total variation pos-
sible among all 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 [L1] and [L2]. They are always
finite even if the total variation of f is infinite, and they may be viewed as
the generalisation of the Hahn-Jordan decomposition for real c\`{a}dl\`{a}g
functions. We also present partial results for more general functions.
