  This paper considers a certain doubly singular parabolic equations with one
singularity occurs in the time derivative, whose model is \begin{equation*}
\partial_t\beta(u)-\operatorname{div}|Du|^{p-2}Du\ni0,\qquad \text{in}\quad
\Omega\times(0,T)\end{equation*} where $\Omega\subset\mathbb{R}^N$ and
$N\geq3$. We show that the bounded weak solutions are locally continuous in the
range $$2-\epsilon_0\leq p<2,$$ provided $\epsilon_0>0$ is small enough, and
the continuity is stable as $p\to2$.
