It is known that the concepts of continuous norm and of absolutely continuous norm do not coincide. There exists a space in which all functions possess continuous norm but not all functions possess absolutely continuous norm. In this paper we construct an extremal example of a Banach function space in which all functions have continuous norm but only the zero function has absolutely continuous norm.
"Continuous Norms and Absolutely Continuous Norms in Banach Function Spaces are not the Same." Real Anal. Exchange 26 (1) 345 - 364, 2000/2001.