In this paper, we improve Caristi-Jachymski-Stein Jr and Banach-Caristi type
fixed point theorems by relaxing the strong continuity assumption of the
mapping with some weaker continuity notions. As an application, we show that
the weaker version of the Caristi-Jachymski-Stein Jr fixed point theorem
characterizes the completeness of the metric space and the Cantor
intersection property.