# converse to Taylor’s theorem

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

###### Theorem.

Let $f\colon U\to{\mathbb{R}}$ be a function such that there exists a constant $C>0$ and an integer $k\geq 0$ such that for each $x\in U$ there is a polynomial $p_{x}(y)$ of $k$ where

 $\lvert f(x+y)-p_{x}(y)\rvert\leq C\lvert y\rvert^{k+1}$

for $y$ near 0. Then $f\in C^{k}(U)$ ($f$ is $k$ continuously differentiable) and the Taylor expansion (http://planetmath.org/TaylorSeries) of $k$ of $f$ about any $x\in U$ 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 ConverseToTaylorsTheorem 2013-03-22 15:05:42 2013-03-22 15:05:42 jirka (4157) jirka (4157) 6 jirka (4157) Theorem msc 41A58 Taylor’s theorem converse TaylorSeries BorelLemma