12

What does this expression mean?

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

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

Jonas Meyer
  • 53,602
Tomas Sironi
  • 271
  • 3
    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 05:20

2 Answers2

17

$f∈C^2[a,b]$ means that $f : [a,b] \rightarrow \mathbb{R}$ is a function that is twice differentiable with each derivative continuous. That is, $f'$ and $f''$ both exist and are both continuous.

$C^0$ means the function is continuous, $C^1$ means the first derivative is continuous, $C^2$ means the second derivative is continuous, In general, $C^n$ means the $n^{th}$ derivative is continuous.

Nearoo
  • 123
  • 5
Hans Parshall
  • 6,246
  • 5
    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 00:25
  • 2
    $C^0$ 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. –  Nov 26 '10 at 00:44
  • 1
    @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 00:46
3

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

TCL
  • 14,262