  In 1934, Whitney raised the question of how to recognize whether a function f
defined on a closed subset X of Euclidean space is the restriction of a
function that is continuously differentiable to order p. A necessary and
sufficient criterion was given in the case n=1 by Whitney, using limits of
finite differences, and in the case p=1 by Glaeser (1958), using limits of
secants. We introduce a necessary geometric criterion, for general n and p,
involving limits of finite differences, that we conjecture is sufficient at
least if X has a "tame topology". We prove that, if X is a compact subanalytic
set, then there exists q=q(p) such that the criterion of order q implies that f
is p times continuously differentiable. The result gives a new approach to
higher-order tangent bundles (or bundles of differentiable operators) on
singular spaces.
