Let

$X=\left(X\right({\Omega}),\parallel \xb7\parallel )$ be a Banach function space and let

$f\in X$ be an arbitrary function. We say that

$f$ has an absolutely continuous norm in

$X$ if for any sequence of open sets

${G}_{n}$,

${G}_{1}\supset {G}_{2}\supset {G}_{3}\supset \cdots $,

${\bigcap}_{n=1}^{\infty}{G}_{n}=\varnothing $, the norms

$\parallel f{\chi}_{{G}_{n}}\parallel $ tend to zero as

$n\to \infty $. We say that

$f$ has the continuous norm in

$X$ if

${lim}_{r\to 0+}\parallel f{\chi}_{(x-r,x+r)\cap {\Omega}}\parallel =0$ for every

$x\in \overline{{\Omega}}$ and

${lim}_{r\to \infty}\parallel f{\chi}_{{\Omega}\setminus (-r,r)}\parallel =0$. Denote the set of all functions with the continuous norm by

${X}_{c}$. The concept of absolutely continuous norm plays a very important role in characterization of classes of reflexive Banach function spaces and of separable Banach function spaces. For instance, it is known that a Banach function space

$X$ is reflexive if and only if both

$X$ and its associate space

${X}^{\text{'}}$ have absolutely continuous norm. The concept of continuous norm was introduced by

*Q. Lai* and

*L. Pick* in [J. London Math. Soc. 48, No. 1, 167-177 (1993;

Zbl 0739.47024)]. They proved that the Hardy operator

$Tf\left(x\right)={\int}_{0}^{x}f\left(t\right)dt$ is a compact mapping from a Banach function space

$(X,v)$ into

${L}^{\infty}$ if and only if the function

$1/v$ has a continuous norm in the associate space

$({X}^{\text{'}},v)$. The authors consider the generalized Lebesgue space

${L}^{p\left(x\right)}$ of functions integrable with variable

$p$ and study relations between its subspaces of functions with continuous norms, subspaces of functions with absolutely continuous norms and subspaces of bounded functions.