Continuity of Hardy-Littlewood Maximal Function

In the paper, we characterize a necessary and sufficient condition which ensures the continuities of the non-centered Hardy-Littlewood maximal function Mf and the centered Hardy-Littlewood maximal function M c f on ℝ n . As two applications, we can easily deduce that M c f and Mf are continuous if f is continuous, and Mf is continuous if f is of local bounded variation on ℝ.