![what is the meaning of $C^0[0,1]$ , $C[0,1]$ and $C^1[0,1]$?](https://liberty.io/uploads/what-is-the-meaning-of-c-0-0-1-c-0-1-and-c-1-0-1_720.jpg)
$\begingroup$Some difficulty to understand the symbol
what is the meaning of $C^0[0,1]$ , $C[0,1]$ and $C^1[0,1]$ ?
$\endgroup$22 Answers
$\begingroup$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 $I$.
$\endgroup$$\begingroup$$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)
$\endgroup$
