# Sard’s theorem

###### Theorem.

Let $U$ be an open set in $R^{m}$, $V$ be an open set in $R^{n}$ and $f:U\to V$ be a $C^{k}$ function. If $k\geq m/n$, then the set of critical values of $f$ has $n$-dimensional measure zero.

## References

• 1 S.G. Krantz, H. R. Parks, The Implicit Function Theorem: History, Theory, and Applications, Birkhäuser, Boston, c. 2002
• 2 H. Federer, Geometric Measure Theory, Springer-Verlag, New York, 1969
Title Sard’s theorem SardsTheorem 2013-03-22 16:12:33 2013-03-22 16:12:33 Mathprof (13753) Mathprof (13753) 7 Mathprof (13753) Theorem msc 57R35