Sard's theorem
/ SARDZ /
The regular-value theorem is wonderful but useless if regular values are rare. Sard's theorem reassures us they are everywhere: for any smooth map, the bad values — the images of points where the differential drops rank — form a negligibly small set. So a regular value is not a lucky accident; almost every value is regular, and you can always perturb to find one.
Precisely, let f: M -> N be smooth (C^infinity, or C^k with k large enough relative to the dimensions). A critical point is a p where df_p is not surjective; a critical value is the image of a critical point. Sard's theorem states that the set of critical values has measure zero in N. Its complement, the regular values, is therefore of full measure and in particular dense; by Baire's theorem it is even residual when f is proper. Note the asymmetry: the set of critical points can be huge — for a constant map every point is critical — but their images, the critical values, are still measure zero.
This is the genericity statement that powers differential topology. It guarantees that level sets f^{-1}(c) are manifolds for almost every c, that a generic smooth map can be perturbed to be transverse to a given submanifold, that Morse functions are dense (almost every height function on an embedded manifold is Morse), and it underlies the well-definedness of degree theory. The honest fine print: measure zero does not mean small in every sense — the regular values are dense but the critical values can also be dense, and the differentiability hypothesis genuinely matters. There are C^1 maps for which the conclusion fails; the required smoothness grows with the dimension drop, which is why one states Sard for C^infinity to be safe.
Project the torus standing upright in R^3 onto the vertical axis as a height function. Its four critical points (bottom, two saddles, top) give four critical heights — a set of measure zero in R — so every other height is a regular value and slices the torus into smooth level curves.
Only finitely many heights are critical; almost all are regular.
Measure zero is not the same as small or closed: critical values can be dense. And the theorem needs enough smoothness — it can fail for merely C^1 maps when the dimension drop is large, so state it for C^infinity.