converse to Taylor’s theorem

Let Un be an open set.


Let f:UR be a function such that there exists a constant C>0 and an integer k0 such that for each xU there is a polynomial px(y) of k where


for y near 0. Then fCk(U) (f is k continuously differentiable) and the Taylor expansionMathworldPlanetmath ( of k of f about any xU is given by px.

Note that when k=0 the hypothesis of the theorem is just that f is LipschitzPlanetmathPlanetmath in U which certainly makes it continuousMathworldPlanetmath in U.


  • 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