what is meant by $ f ∈ C^{2}[a, b] ?$

Question

What is the meaning of $ f ∈ C^{2}[a, b] ?$
I think it says that $f$ is twice differential on $[a,b]$, isn't it?

Answer 1

In general, if $f\in C^k,$ then the derivatives $f',\ f'',\ \dots,\ f^{(k)}$ exist and are continuous. Here is the reference.

Answer 2

It means that $f^{\prime\prime}$ is continuous on $[a,b]$.

Answer 3

$f$ is twice differentiable on $\left [a, b \right]$ and $f''$ is continuous on $\left [a,b \right]$. The derivatives at the boundaries are to be understood in the sense of one-sided derivatives.