# What does “$f\in C^2[a,b]$” mean?

What does this expression mean?

$$f\in C^2[a,b]$$

More specifically, I don't know what $C$ means.

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as has been noted, the "$C$" stands for "continuous", the $2$ for "second derivative"; so functions with domain $[a,b]$ that have second derivatives that are continuous. – Arturo Magidin Nov 26 '10 at 5:20

## 2 Answers

$f : [a,b] \rightarrow \mathbb{R}$ is a function that is twice differentiable with each derivative continuous. That is, $f'$ and $f''$ both exist and are both continuous.

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Did you mean $f:[a,b] \rightarrow \mathbb{R}$? – Shai Covo Nov 25 '10 at 20:44
In other words, $C^2$ means that the function and its first derivative is continuous in $[a,b]$? – Tomas Sironi Nov 26 '10 at 0:25
$C^0$ means the function is continuos, $C^1$ means the first derivative is continuos, $C^2$ means the second derivative is continuos, In general, $C^{n}$ means the $n^{th}$ derivative is continuos. – user17762 Nov 26 '10 at 0:44
@tomm89: It means that $f$, $f'$, and $f''$ are continuous on $[a,b]$ (equivalently, $f''$ is continuous on $[a,b]$, as Sivaram noted). – Shai Covo Nov 26 '10 at 0:46
@Shai: yes, thank you. Fixed. – Hans Parshall Nov 26 '10 at 1:21

$f:[a,b]\to \mathbb{R}$ or $\to \mathbb{C}$. And both the first and second derivatives of $f$ are continuous.

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