In this paper, we solve the following extension problem.
Problem 1. Suppose we are given a function f : E → R, where E is a
given subset of Rn. How can we decide whether f extends to a Cm−1,1 function
F on Rn ?
Here, m ≥ 1 is given. As usual, Cm−1,1 denotes the space of functions
whose (m − 1)rst derivatives are Lipschitz 1. We make no assumption on the
set E or the function f.
This problem, with Cm in place of Cm−1,1, goes back to Whitney ,
, . To answer it, we prove the following sharp form of the Whitney