Function as element of a set

Question

Can anyone tell me what does the following notation represent $$f \in C^2[a,b]$$ in terms of function notation?

I suppose $[a,b]$ means the function is defined in that interval, but I am confused with the '$f \in$' notation.

Answer 1

$C^n[a,b]$ is the set of all functions defined in $[a,b]$ that have up to n-order continuous derivatives. So the notation means that both $f'$ and $f''$ exist and are continuous.