Level set
From Encyclopedia of Mathematics
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.
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://encyclopediaofmath.org/index.php?title=Level_set&oldid=41809
Level set. Encyclopedia of Mathematics. URL: http://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