What does "$f\in C^2[a,b]$" mean?

$f∈C^2[a,b]$ means that $f : [a,b] \rightarrow \mathbb{R}$ is a function that is twice differentiable with each derivative continuous. That is, $f'$ and $f''$ both exist and are both continuous.

$C^0$ means the function is continuous, $C^1$ means the first derivative is continuous, $C^2$ means the second derivative is continuous, In general, $C^n$ means the $n^{th}$ derivative is continuous.

$f:[a,b]\to \mathbb{R}$ or $\to \mathbb{C}$. And both the first and second derivatives of $f$ are continuous.