  We report the detailed study of the irreversibility field {\hirr} of single
crystals of Mg(B$_{1-x}$C$_{x}$)$_2$ ($x$=0, 2, 3.5, 5 and 7%) in strong
magnetic fields of up to 40 T. High sensitive torque measurements revealed that
the {\hirr} is greatly enhanced by a factor of two
($\mu_{0}${\hirr}(0)$\simeq$33 T in $x$=5%) by carbon substitution compared to
the pristine {\mgb} owing to a reduction in the inplane coherence length. We
also observed the temperature-dependent {\hirr} anisotropy in all the carbon
contents. This fact strongly suggests that the two-gap superconductivity of
{\mgb} is less influenced by a small amount of carbon substitution. We discuss
these results focusing on the impact of carrier doping and impurity scattering
on two-gap superconductor.
