How to read $g \in C^2 (\Omega \times \mathbb{R}; \mathbb{R})$

Question

I came across the following litteral describing a function $g$ mapping from a two dimensional domain $\Omega$ into the real numbers $\mathbb{R}$: $ g \in C^2 (\Omega \times \mathbb{R}; \mathbb{R}) $

How do I read it correctly, especially, what's describing the "$\in C^2$ " here?

Answer 1

If you want to know just how to read : $g$ is continuously twice differentiable map from $\Omega\times\mathbb{R}$ to $\mathbb{R}$.

If you don’t know what is $C^2$ : When a function is in class of $C^n$, it means this function is $n$ times continuously differentiable within the given domain. Also, $C^0$ is used to note the class of continuous functions. For example, by writing $f\in C^1(A;B)$, it means $f$ is a map from $A$ to $B$, which is continuously differentiable on $A$.