Lately I have been running into the following notation fairly often: $f\in C^0(\Omega)$, $g\in C^0(B,\mathbb R^3)$ etc.
I always thought this was just some peoples' way of saying that a function is simply continuous, but lately I am confused about this notation. In the same text the author repeatedly mixes the notation $f\in C(\Omega)$ and $u\in C^0(\Omega)$.
Does this mean the same? What is the difference between these notations?
Usually, the notations mean the same.
One tends to use $C^0(\Omega)$ rather than $C(\Omega)$ if/when one also uses $C^k(\Omega)$ spaces with $k>0$ at the same time or at least in close proximity, for the uniformity of notation. But it's also not uncommon to stick to $C(\Omega)$ for the space of continuous functions even then.
In contexts where $C^k(\Omega)$ doesn't make sense (if $\Omega$ is an arbitrary topological space, not something similar to a manifold), $C^0(\Omega)$ is very rarely used if at all.
If the author mixes notations, it's probably just inconsistent notation, but it might be that the author denotes different spaces by the different symbols. If that is the case, the author ought to have stated that somewhere conspicuous, since the two notations have the same standard meaning.
