  In this paper we address the $W^{1,1}$-continuity of several maximal
operators at the gradient level. A key idea in our global strategy is the
decomposition of a maximal operator, with the absence of strict local maxima in
the disconnecting set, into "lateral" maximal operators with good monotonicity
and convergence properties. This construction is inspired in the classical
sunrise lemma in harmonic analysis. A model case for our sunrise strategy
considers the uncentered Hardy-Littlewood maximal operator $\widetilde{M}$
acting on $W^{1,1}_{\rm rad}(\mathbb{R}^d)$, the subspace of
$W^{1,1}(\mathbb{R}^d)$ consisting of radial functions. In dimension $d\geq 2$
it was recently established by H. Luiro that the map $f \mapsto \nabla
\widetilde{M} f$ is bounded from $W^{1,1}_{\rm rad}(\mathbb{R}^d)$ to
$L^1(\mathbb{R}^d)$, and we show that such map is also continuous. Further
applications of the sunrise strategy in connection with the
$W^{1,1}$-continuity problem include non-tangential maximal operators on
$\mathbb{R}^d$ acting on radial functions when $d\geq 2$ and general functions
when $d=1$, and the uncentered Hardy-Littlewood maximal operator on the sphere
$\mathbb{S}^d$ acting on polar functions when $d\geq 2$ and general functions
when $d=1$.
