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We prove the continuity of the map $f \mapsto (Mf)'$ from$W^{1,1}(\mathbb{R})$ to $L^1(\mathbb{R})$, where $M$ is the centeredHardy-Littlewood maximal operator. This solves a question posed by Carneiro,Madrid and Pierce.

Continuity for the one-dimensional centered Hardy-Littlewood maximal operator at the derivative level