TITLE: Basic Question about notation in the space of continuous functions
QUESTION [5 upvotes]: I am reading the book "Introduction to Calculus of Variations" by Bernard Dacorogna (Could not find a link in google books) where he defines
$C(\bar{\Omega})$ to be the space of continuous functions $u : \Omega \to \mathbb{R}$ which can be continuously extended to $\bar{\Omega}$ (here $\Omega$
is an open set in $\mathbb{R}^n$).
After this, he defines the norm over $C(\bar{\Omega})$ by 
$\|u\|_0 = \sup_{x \in \bar{\Omega}} |u(x)|$ and then $C(\bar{\Omega})$ with this norm is a Banach Space.
I am confused about two things:-
1) Does $C(\bar{\Omega})$ consist of functions $u : \Omega \to \mathbb{R}$ which can be continuously extended or functions $u : \bar{\Omega} \to \mathbb{R}$
2)The "norm" $\|.\|_0$ is not a norm as $\bar{\Omega}$ need not be bounded and
hence it can take an infinite value
Can someone please let me know if I am right and if so, this notation is OK in some texts.

REPLY [5 votes]: I included a screenshot below. The concept of $C(\overline{\Omega})$ is unambigious, as user161825 pointed out. If a continuous extension to the closure exists, it is unique and we may consider the function as already  extended. 
Item 2) is a mess-up on the author's part. Just assume $\Omega$ is a bounded open set, because it will be practically everywhere in the book. 
I might add that this is not a book I would recommend for learning the calculus of variations. The variational chapters in Brezis and Evans would be a better source, partly because they are better written, partly because they are not laser-beam focused on the Italian tradition of the subject.