Sard's theorem

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This article gives the statement and possibly proof of a theorem that discusses regular values, critical values, regular points or critical points of a smooth map between differential manifolds

Statement

Suppose M and N are differential manifolds and f:M \to N is a smooth map between them. Then, the set of regular values of f is a subset of measure zero in N.

Applications

Suppose M and N are differential manifolds, and the dimension of M is strictly less than the dimension of N. Then, if f:M \to N is a smooth map, the image f(M) has measure zero as a subset of N. In particular, f cannot be surjective.

This also shows that a differential manifold cannot be expressed as a union of the images of countably many smooth maps from differential manifolds of strictly smaller dimension