On the variation of the Hardy-Littlewood maximal function

We show that a function $f$ of bounded variation satisfies $$\Var Mf \leq C \Var f$$ where $Mf$ is the centered Hardy-Littlewood maximal function of $f$. Consequently, the operator $f \mapsto (Mf)'$ is bounded from $W^{1,1}(R)$ to $L^{1}(R)$. This answers a question of Hajlasz and Onninen in the one-dimensional case.