Symbol interpretation (Non-maths background)

Question

Please explain the symbols used below the red line and how to interpret it? Thanks.

Answer 1

$C^2(\mathbb{R}^2, \mathbb{R})$ is the set of twice continuously differentiable real valued functions of two real variables.

In general, $C^n(A,B)$ is the set of $n$ times continuously differentiable functions with domain $A$ and codomain $B$. (That makes sense only when differentiation makes sense for functions from $A$ to $B$.)

When $n=0$ it's the set of continuous functions. Then the $0$ is often omitted.

Answer 2

If $f(w,t)$ is a function from $\mathbb R^2$ to $\mathbb R$ that has a second derivative that is continuous....

Answer 3

It means $f(w,t)$ is a real-valued function that takes two variables ($w$ and $t$) and is twice-differentiable, and the second-order partial derivatives are continuous.