# converse to Taylor’s theorem

Let $U\subset {\mathbb{R}}^{n}$ be an open set.

###### Theorem.

Let $f\mathrm{:}U\mathrm{\to}\mathrm{R}$ be a function such that there exists a constant $C\mathrm{>}\mathrm{0}$ and an integer $k\mathrm{\ge}\mathrm{0}$ such that for each $x\mathrm{\in}U$ there is a polynomial ${p}_{x}\mathit{}\mathrm{(}y\mathrm{)}$ of $k$ where

$$|f(x+y)-{p}_{x}(y)|\le C{|y|}^{k+1}$$ |

for $y$ near 0. Then $f\mathrm{\in}{C}^{k}\mathit{}\mathrm{(}U\mathrm{)}$ ($f$ is $k$ continuously differentiable) and the Taylor expansion^{} (http://planetmath.org/TaylorSeries) of $k$ of $f$ about any $x\mathrm{\in}U$ is given by ${p}_{x}$.

Note that when $k=0$ the hypothesis of the theorem is just that $f$ is Lipschitz^{} in $U$ which certainly makes it continuous^{} in $U$.

## References

- 1 Steven G. Krantz, Harold R. Parks. . Birkhäuser, Boston, 2002.

Title | converse to Taylor’s theorem |
---|---|

Canonical name | ConverseToTaylorsTheorem |

Date of creation | 2013-03-22 15:05:42 |

Last modified on | 2013-03-22 15:05:42 |

Owner | jirka (4157) |

Last modified by | jirka (4157) |

Numerical id | 6 |

Author | jirka (4157) |

Entry type | Theorem |

Classification | msc 41A58 |

Synonym | Taylor’s theorem converse |

Related topic | TaylorSeries |

Related topic | BorelLemma |