Sard's theorem

Statement

More explicitly, let

${\displaystyle f:{\mathbb{R}}^n\to {\mathbb{R}}^m}$

be ${\displaystyle C^k}$, (that is, ${\displaystyle k}$ times continuously differentiable), where ${\displaystyle k\ge max\{n-m+1,1\}}$. Let ${\displaystyle X\subset {\mathbb{R}}^n}$ denote the critical set of ${\displaystyle f,}$ which is the set of points ${\displaystyle x\in {\mathbb{R}}^n}$ at which the Jacobian matrix of ${\displaystyle f}$ has rank ${\displaystyle <m}$. Then the image ${\displaystyle f(X)}$ has Lebesgue measure 0 in ${\displaystyle {\mathbb{R}}^m}$.

Intuitively speaking, this means that although ${\displaystyle X}$ may be large, its image must be small in the sense of Lebesgue measure: while ${\displaystyle f}$ may have many critical points in the domain ${\displaystyle {\mathbb{R}}^n}$, it must have few critical values in the image ${\displaystyle {\mathbb{R}}^m}$.

More generally, the result also holds for mappings between differentiable manifolds ${\displaystyle M}$ and ${\displaystyle N}$ of dimensions ${\displaystyle m}$ and ${\displaystyle n}$, respectively. The critical set ${\displaystyle X}$ of a ${\displaystyle C^k}$ function

${\displaystyle f:N\to M}$

consists of those points at which the differential

${\displaystyle df:TN\to TM}$

has rank less than ${\displaystyle m}$ as a linear transformation. If ${\displaystyle k\ge max\{n-m+1,1\}}$, then Sard's theorem asserts that the image of ${\displaystyle X}$ has measure zero as a subset of ${\displaystyle M}$. This formulation of the result follows from the version for Euclidean spaces by taking a countable set of coordinate patches. The conclusion of the theorem is a local statement, since a countable union of sets of measure zero is a set of measure zero, and the property of a subset of a coordinate patch having zero measure is invariant under diffeomorphism.