Is this an inconsistence in the usage of the space of all continuously differentiable functions $C^{n}(X, E)$?

Question

My textbook defines the space of all continuously differentiable functions $C^{n}(X, E)$ as follows:

As such, $f \in C^{n}(X, E)$ IFF the $n$-th derivative of $f$ is continuous on $X$. I found that this definition coincides with one given by Wikipedia's page.

Roll's theorem is stated in my textbook as:

From the definition, I got that $f \in C([a, b], \mathbb{R})$ IFF $f'$ is continuous on $[a,b]$, which is stronger than "$f$ is differentiable on $(a,b)$". This causes me confusion.

The authors already have $f \in C([a, b], \mathbb{R})$, but they add that $f$ is differentiable on $(a,b)$. Is it a redundancy?

Please elaborate on this notion!

Answer 1

You're confusing $C^1$ with $C$. $C$ just means $f$ is continuous and places no requirements on differentiability. $C^1$ means differentiable with continuous derivative.

Answer 2

$C$ is $C^{0}$ and the zeroth derivative of a function is the function itself.