Here is a definition of $C_0(\Omega)$ in wikipedia: (http://en.m.wikipedia.org/wiki/Vanish_at_infinity)
Let $(X,\tau)$ be a locally compact space.
Let's call "a function $f:X\rightarrow \mathbb{F}$ vanishes at infinity" iff $\forall \epsilon >0, \exists$ a compact subset $K$ of $X$ such that $\forall x\in X\setminus K, |f(x)|<\epsilon$.
Then $C_0(X,\mathbb{F})$ is defined as the set of functions $f:X\rightarrow \mathbb{F}$ vanish at infinity.
As it says, this definition does not require functions to be continuous, and it's really weird since one usually denotes set of functions as '$C$' when it is a subset of the set of continuous functions..
Also, I have seen some papers require $C_0$ to be continuous.
What is the usual definition of $C_0(X)$?