Let $\phi : X^n \rightarrow Y^m$ be a smooth map on smooth manifolds. A critical point of $\phi$ is a point $p\in X$ such that the differential $\phi_* : T_pX \rightarrow T_{\phi(p)}Y$ considered as a linear transformation of real vector spaces has rank $<m$ . A critical value of $\phi$ is the image of a critical point. A regular value of $\phi$ is a point $q\in Y$ which is not the image of any critical point. In particular, $q$ is a regular value of $\phi$ if $q\in Y \setminus \phi(X)$ .

Following Spivak [Spivak], we say a subset $V$ of $Y^m$ has measure zero if there is a sequence of coordinate charts $(x_i,U_i)$ whose union contains $V$ and such that $x_i(U_i\cap V)$ has measure 0 (in the usual sense) in $\mathbb{R}^m$ for all $i$ . With that definition, we can now state:

Sard's Theorem. Let $\phi : X \rightarrow Y$ be a smooth map on smooth manifolds. Then the set of critical values of $\phi$ has measure zero.

Bibliography

Spivak

Spivak, Michael. A Comprehensive Introduction to Differential Geometry. Volume I, Third Edition. Publish of Perish, Inc. Houston, Texas. 1999.