what is the meaning of $C^0[0,1]$ , $C[0,1]$ and $C^1[0,1]$?

Question

Some difficulty to understand the symbol

what is the meaning of $C^0[0,1]$ , $C[0,1]$ and $C^1[0,1]$ ?

Answer 1

For $n\ge 1, \; C^n(I) \;$ is the set of functions for which the $n^{\text{ th}}$ derivative $f^{(n)}$ is well defined at the intervall $I$ and is Continuous (at $I$).

for $n=0$,

$C^0(I)=C(I)$ contains all continuous functions at $I$.

Answer 2

$C[0,1] = C^0[0,1]$ is the set of functions that are continuous over the interval $[0,1]$.

$C^1[0,1]$ is the set of functions that are derivable over $[0,1]$ and which derivatives are continuous as well.

$C^n[0,1]$, for $n\in \mathbb{N}_0$, is the set of functions that are derivable (at least) $n$ times and where the $n$. derivative is in $C[0,1]$.

(Note functions in $C^1[0,1]$ are continuous since they are derivable)