  Using a simple solvable model, i.e., Higgs--Yukawa system with an infinite
number of flavors, we explicitly demonstrate how a dimensional continuation of
the $\beta$ function in two dimensional MS scheme {\it fails\/} to reproduce
the correct behavior of the $\beta$ function in four dimensions. The mapping
between coupling constants in two dimensional MS scheme and a conventional
scheme in the cutoff regularization, in which the dimensional continuation of
the $\beta$ function is smooth, becomes singular when the dimension of
spacetime approaches to four. The existence of a non-trivial fixed point in
$2+\epsilon$ dimensions continued to four dimensions $\epsilon\to2$ in the two
dimensional MS scheme is spurious and the asymptotic safety cannot be imposed
to this model in four dimensions.
