converse to Taylor’s theorem

Let $U\subset {ℝ}^n$ 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 $U$.