View source for Sard theorem

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083230/s0832301.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083230/s0832302.png" />-mapping of manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083230/s0832303.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083230/s0832304.png" /> of dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083230/s0832305.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083230/s0832306.png" />, respectively; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083230/s0832307.png" />, then the critical values (cf. [[Critical value|Critical value]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083230/s0832308.png" /> form a set of measure zero. The set of regular values turns out to be of full measure and everywhere dense. The theorem was proved by A. Sard [[#References|[1]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Sard, "The measure of critical values of differentiable maps" ''Bull. Amer. Math. Soc.'' , '''48''' (1942) pp. 883–890 {{MR|7523}} {{ZBL|0063.06720}} </TD></TR></table>



====Comments====
"Full measure" is, in the Russian article, called "massive setmassive" . See also [[Singularities of differentiable mappings|Singularities of differentiable mappings]].