Graphs and Level Sets

TL;DR: In this paper, the authors define a set of sets, called level sets, which are useful in studying qualitative properties of a real valued function of several real variables, and describe them as the solution set of the equation f(x1,x n + 1) = c.

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.”