  We prove some new continuity results for the Julia sets $J$ and $J^{+}$ of
the complex H\'enon map $H_{c,a}(x,y)=(x^{2}+c+ay, ax)$, where $a$ and $c$ are
complex parameters. We look at the parameter space of dissipative H\'enon maps
which have a fixed point with one eigenvalue $(1+t)\lambda$, where $\lambda$ is
a root of unity and $t$ is real and small in absolute value. These maps have a
semi-parabolic fixed point when $t$ is $0$, and we use the techniques that we
have developed in [RT] for the semi-parabolic case to describe nearby
perturbations. We show that for small nonzero $|t|$, the H\'enon map is
hyperbolic and has connected Julia set. We prove that the Julia sets $J$ and
$J^{+}$ depend continuously on the parameters as $t\rightarrow 0$, which is a
two-dimensional analogue of radial convergence from one-dimensional dynamics.
Moreover, we prove that this family of H\'enon maps is stable on $J$ and
$J^{+}$ when $t$ is nonnegative.
