  We investigate dynamical chiral symmetry breaking in vector-like gauge
theories in $D$ dimensions with ($D-4$) compactified extra dimensions, based on
the gap equation (Schwinger-Dyson equation) and the effective potential for the
bulk gauge theories within the improved ladder approximation. The non-local
gauge fixing method is adopted so as to keep the ladder approximation
consistent with the Ward-Takahashi identities.
  Using the one-loop $\bar{\rm MS}$ gauge coupling of the truncated KK
effective theory which has a nontrivial ultraviolet fixed point (UV-FP) $g_*$
for the (dimensionless) bulk gauge coupling ${\hat g}$, we find that there
exists a critical number of flavors, $N_f^{\rm crit}$ ($\simeq 4.2, 1.8$ for
$D=6, 8$ for SU(3) gauge theory): For $N_f > N_f^{\rm crit}$, the dynamical
chiral symmetry breaking takes place not only in the ``strong-coupling phase''
(${\hat g} >g_*$) but also in the ``weak-coupling phase'' (${\hat g} <g_*$)
when the cutoff is large enough. For $N_f < N_f^{\rm crit}$, on the other hand,
only the strong-coupling phase is a broken phase and we can formally define a
continuum (infinite cutoff) limit, so that the physics is insensitive to the
cutoff in this case.
  We also perform a similar analysis using the one-loop ``effective gauge
coupling''. We find the $N_f^{\rm crit}$ turns out to be a value similar to
that of the $\bar{\rm MS}$ case, notwithstanding the enhancement of the
coupling compared with that of the $\bar{\rm MS}$.
