  We prove that the set of solutions to the parabolic singular $p$-Laplace
equation with Dirichlet boundary conditions on a bounded Lipschitz domain
$\Omega$ for all space dimensions is continuous in the parameter $p\in
[1,+\infty)$ and the initial data. The highly singular limit case p=1 is
included. In particular, we show that the solutions $u_p$ converge strongly in
$L^2(\Omega)$, uniformly in time, to the solution $u_1$ of the parabolic
1-Laplace equation as $p\to 1$.
