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Will you provide a simple application of Sard's theorem using some specific functions?

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Sard's theorem is not usually applied to specific known functions, but rather to some function you know only a little about, but which you can at least conclude has a regular value using Sard's theorem. In topology one often uses it show that you can define a manifold as the preimage of a regular value. –  Grumpy Parsnip Mar 21 '11 at 20:09
    
Dear Thales, As another example of Jim's comment, Sard's theorem gives the generic smoothness theorem in algebraic geometry over $\mathbb{C}$ (and thus in characteristic zero by the Lefschetz principle, at least for quasi-projective varieties). –  Akhil Mathew Mar 22 '11 at 1:28
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up vote 3 down vote accepted

as noted in the comments sard's theorem isn't really used in this way, but here is an example of the sort of thing it is about. for $f:\mathbb{R}^2\to\mathbb{R}, f(x,y)=x^2+y^2$ we have the derivative zero only at the origin (the set of critical points), and $f(0,0)$ (the set of critical values) is just a single point, which has measure zero in $\mathbb{R}$. for any other $r>0$, $f^{-1}(r)$ is the circle of radius $r$, a manifold of dimension $1=2-1$. the theorem is used often to say you can find (lots of) points like $r\in(0\infty)$ that have nice preimages.

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