Graphs and Level Sets

Abstract

Associated with each real valued function of several real variables is a collection of sets, callcd level sets, which are useful in studying qualitative properties of the function. Given a function f: U → ℝ. where U ⊂ ℝ^{n +1},its level sets are the sets f^-1 (c) defined, for each real number c, by

$$ {f^{ - 1}}\,\left( c \right)\, = \,\left\{ {\left( {{X_1},...,\,{X_{n + 1}}} \right)\, \in U:\,f\,\left( {{X_1},...,\,{X_{n + 1}}} \right)\, = \;c} \right\} $$

. The number c is called the height of the level set, and f^-1(c) is called the level set at height c. Since f^-l(c) is the solution set of the equation f(x₁…, x _{n +1}) = c, the level set f^-1(c) is often described as “the set f(x₁,…, x _{n +1}) = c.”