  In this paper we investigate some questions related to the continuity of
maximal operators in $W^{1,1}$ and $BV$ spaces, complementing some well-known
boundedness results. Letting $\widetilde M$ be the one-dimensional uncentered
Hardy-Littlewood maximal operator, we prove that the map $f \mapsto
\big(\widetilde Mf\big)'$ is continuous from $W^{1,1}(\mathbb{R})$ to
$L^1(\mathbb{R})$. In the discrete setting, we prove that $\widetilde M:
BV(\mathbb{Z}) \to BV(\mathbb{Z})$ is also continuous. For the one-dimensional
fractional Hardy-Littlewood maximal operator, we prove by means of
counterexamples that the corresponding continuity statements do not hold, both
in the continuous and discrete settings, and for the centered and uncentered
versions.
