Abstract: This paper focuses on the continuity of the truncated Hardy-Littlewood maximal function. We first show that the truncated Hardy-Littlewood maximal function is lower semi-continuous. Then by investigating the behavior of the truncated Hardy-Littlewood maximal function when the truncated parameter γ changes, we obtain an equivalent condition of the continuity of the truncated Hardy-Littlewood maximal function.

Key words: Hardy-Littlewood maximal function, truncation, continuity, truncated parameter