Vol. 36, No. 2, 1971

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A constructive proof of Sard’s theorem

Yuen-Kwok Chan

Vol. 36 (1971), No. 2, 291–301

The theorem of Sard asserts that if a mapping F from a region in R7n to Rp is smooth enough, then the set of critical values of F has measure zero in Rp. The purpose of this paper is to give a constructive proof of this theorem. By a constructive proof is meant one which has numerical content, as explained in E. Bishop’s Foundations of Constructive Analysis. In particular, it is shown that in every open ball in Rp one can compute a point which is not a critical value of F.

The proof is based on one given by Milnor, which is a modification of a proof of Pontryagin. These proofs, as well as all other known proofs, are nonconstructive, and it is not obvious that they can be constructivized. One difficulty lies in the fact that, given two real numbers a and b, one cannot, in general, prove constructively that either α b or α < b; one can only prove, for arbitrary 𝜖 > 0, that either a > b 𝜖 or a < b. This fact forces, among other things, the consideration of ‘nearly critical values’ instead of critical values, and the derivation of a slightly more general result. Once a proper interpretation for “nearly critical values” has been found, Milnor’s proof can be followed, replacing various nonconstructive arguments by constructive ones.

Mathematical Subject Classification
Primary: 57.20
Secondary: 02.00
Received: 9 March 1970
Published: 1 February 1971
Yuen-Kwok Chan