# Is every function from $W^{1,1}([0,1])$ continuous?

I am starting to learn about Sobolev spaces. I am trying to understand first a bit better what $$W^{1,1}(I)$$ is where $$I=[0,1]$$. I was wondering if every function in this space has to be continuous.

• For every $u \in W^{1,p}$ there exists an absolute continuous function $\hat u \colon I \to \overline {\mathbb R}$ such that $u=\hat u$ almost everywhere Commented Nov 30, 2020 at 7:22
• I see. Then since elements of $W^{1,1}(I)$ are in particular functions in $L^1([0,1])$ we can choose a representative that is continuous everywhere right? Commented Nov 30, 2020 at 10:31
• Yes, every function $u \in W^{1,p}$ has a (unique) continuous representative: there exists a continuous function $\hat u$ such that $\hat u$ belongs to the equivalence class of $u$ (under the relation of $z=w$ a.e.) Commented Nov 30, 2020 at 10:53
• okay thanks, I get it now Commented Nov 30, 2020 at 10:57
• @EvangelopoulosPhoevos where can I look that up sir? Commented Aug 25, 2023 at 15:58