Abstract. In 1934, Whitney raised the question of how to recognize whether a function f defined on a closed subset X of R n is the restriction of a function of class C 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 X (p) such that the criterion of order q implies that f is C p . The result gives a new approach to higher-order tangent bundles (or bundles of differentiable operators) on singular spaces.
Whitney [W2] proved that a function defined on a closed subset of R is the restriction of a C m function if the limiting values of all m th divided differences form a continuous function. We show that Fefferman's solution of Whitney's problem for R n [F, Th. 1] is equivalent to a variant of our conjecture in [BMP2] giving a criterion for C m extension in terms of iterated limits of finite differences.
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