What functions are in $C_0(\Omega)$ that are not in $C_c(\Omega)$?

Question

I use $C_0(\Omega)$ and $C_c(\Omega)$ pretty much interchangeably but I'm trying to be more precise now so I'd like to know what is the difference between these spaces?

$C_0$ is commonly referred to as the set of all continuous functions on $\Omega \in \mathbb{R}^n$ that vanish at infinity (or on the boundary).

$C_c(\Omega)$ is the space of all continuous functions on $\Omega$ with compact support.

It holds that $C_c(\Omega) \subset C_0(\Omega)$. So what functions are in $C_0(\Omega)$ that are not in $C_c(\Omega)$?

Answer 1

Remember that $\partial \Omega$ is a closed set. So if $f \in C_c(\Omega)$ has compact support in $\Omega$, the support of $f$ and $\partial \Omega$ have a strictly positive distance from each other. That means that you can fit a little open set between $\Omega$ and the support of $f$.

This is a much stronger condition than simply requiring $f$ tends to zero on the boundary. Being able to have a constraint on the distance between $\Omega^c$ and where $f$ has interesting behavior is frequently important in proving things about $f$.

Answer 2

For example you can consider $f(x)=e^{-||x||}$