I'm reading "Pseudo-differential Operators and the Nash-Moser Theorem" and at the top of on page 8 they write: "Finally, we note that if $u \in C^0(\Omega)$, then the support of $u$ defined above coincides with the closure of $\{x \in \Omega \mid u(x) \neq 0 \}$." $\Omega$ here is an open subset of $\mathbb{R}^n$.
The support of a distribution $u$ was defined on the previous page as "the complement in $\Omega$ of the points in the neighbourhood of which $u$ is zero".
Question 1: What's the difference between $C(\Omega)$ and $C^0(\Omega)$?
Question 2: From $\operatorname{supp}{u} \subset \Omega$ I deduce that we think of $u$ here as a distribution on a subset of reals. How does this make sense? If my test functions are constants then the integral of $cu$ might not be finite.
Thanks for your help.