  A regularity result for free-discontinuity energies defined on the space
$SBV^{p(\cdot)}$ of special functions of bounded variation with variable
exponent is proved, under the assumption of a log-H\"older continuity for the
variable exponent $p(x)$. Our analysis expand on the regularity theory for
minimizers of a class of free-discontinuity problems in the nonstandard growth
case. This may be seen as a follow-up of the paper Fusco, Mingione and
Trombetti (2001), dealing with a constant exponent.
