# ${C}^{n}$

Let $f:\mathbb{R}\to \mathbb{R}$ be a function. We say that $f$ is of class ${C}^{1}$ if ${f}^{\prime}$ exists and is continuous^{}.

We also say that $f$ is of class ${C}^{n}$ if its $n$-th derivative^{} exists and is continuous (and therefore all other previous derivatives exist and are continuous too).

The class of continuous functions is denoted by ${C}^{0}$. So we get the following relationship among these classes:

$${C}^{0}\supset {C}^{1}\supset {C}^{2}\supset {C}^{3}\supset \mathrm{\dots}$$ |

Finally, the class of functions that have continuous derivatives of any order is denoted by ${C}^{\mathrm{\infty}}$ and thus

$${C}^{\mathrm{\infty}}=\bigcap _{n=0}^{\mathrm{\infty}}{C}^{n}.$$ |

It holds that any function that is differentiable^{} is also continuous
(see this entry (http://planetmath.org/DifferentiableFunctionsAreContinuous)).
Therefore, $f\in {C}^{\mathrm{\infty}}$ if and only if every derivative of $f$ exists.

The previous concepts can be extended to functions $f:{\mathbb{R}}^{m}\to \mathbb{R}$,
where $f$ being of class ${C}^{n}$ amounts to asking that all the
partial derivatives^{} of order $n$ be continuous.
For instance, $f:{\mathbb{R}}^{m}\to \mathbb{R}$ being ${C}^{2}$ means that

$$\frac{{\partial}^{2}f}{\partial {x}_{j}\partial {x}_{i}}$$ |

exists and are all continuous for any $i,j$ from $1$ to $m$.

## ${C}^{n}$ functions on an open set of ${\mathbb{R}}^{m}$

Sometimes we need to talk about continuity not globally on $\mathbb{R}$, but on some interval or open set.

If $U\subseteq {\mathbb{R}}^{m}$ is an open set, and $f:U\to \mathbb{R}$ (or $f:U\to \u2102$) we say that $f$ is of class ${C}^{n}$ if ${\partial}^{\alpha}f$ exist and are continuous for all multi-indices $\alpha $ with $|\alpha |\le n$. See this page (http://planetmath.org/MultiIndexNotation) for the multi-index notation.

Title | ${C}^{n}$ |
---|---|

Canonical name | Cn |

Date of creation | 2013-03-22 14:59:43 |

Last modified on | 2013-03-22 14:59:43 |

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 13 |

Author | drini (3) |

Entry type | Definition |

Classification | msc 46G05 |

Classification | msc 26B05 |

Classification | msc 26A99 |

Classification | msc 26A24 |

Classification | msc 26A15 |

Synonym | ${C}^{1}$ |

Synonym | ${C}^{2}$ |

Synonym | ${C}^{k}$ |

Synonym | ${C}^{\mathrm{\infty}}$ |

Related topic | Derivative |

Related topic | SmoothFunctionsWithCompactSupport |