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Level set

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of a function $f$ on $\mathbf{R}^n$

The set of points in $\mathbf{R}^n$ on which $f= \text{const}$. If the function $f$ is given on a square $Q$ of the plane $\mathbf{R}^2$ and has partial derivatives there which also satisfy a Lipschitz condition, then for almost-all $c$ in the interval $\min f \le c \le \max f$ the level set $$ M_c = \{ x \in Q \ :\ f(x) = c \} $$ consists of a finite number of regular curves (on them, $\mathrm{grad}\,f \ne 0$). Cf. Sard theorem.

How to Cite This Entry:
Level set. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Level_set&oldid=41809
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article