What does $C^0$ stand for?
Solution 1:
I made it a post because I am not allowed to comment yet. Usually in the textbook of functional analysis and PDEs we use the following definition: Let $\Omega$ be open subset of $\mathbb{R}$, then
$$C^0 (\overline{\Omega}, \mathbb{R})= \{ f: \Omega \to \mathbb{R}, f \text{ is continuous}\}$$ equipped with the norm
$$\| f\|_{c^0_b} = \sup_{x \in \Omega} | f(x)|.$$
So $C^0$ is just the continuous functions. See this link for the proof: https://math.stackexchange.com/questions/2750127/hölder-continuous-functions-are-uniformly-continuous.