I kept on looking for him outside my building and in the park, but I still haven't found him.

kept 是动词 keep 的过去式，keep 的意思是"保持、继续"。 on 在这里是副词，表示"持续、继续"的状态。 looking 是动词 look 的现在分词形式，look for 是固定搭配，意思是"寻找"。 kept on looking 这个短语表示"持续不断地寻找"，强调动作的持续性。