Level set

Definition

Let $D \subseteq \mathbb{R}^n$, let $f \colon D \to \mathbb{R}$ be a function of several variables, and let $c \in \mathbb{R}$. The level set of $f$ at the value $c$ is the set $$L_c(f) = \{x \in D : f(x) = c\}.$$ It is the preimage $f^{-1}(\{c\})$ of the one-point set $\{c\}$. For $n = 2$ a level set is also called a level curve, and for $n = 3$ it is also called a level surface.

Remarks

For $f(x, y) = x^2 + y^2$, the level set at $c > 0$ is the circle of radius $\sqrt{c}$ about the origin. The level set at $c = 0$ is the point $(0, 0)$, and the level set at $c < 0$ is empty. The level sets of $f$ for different values of $c$ are disjoint, and their union is the domain $D$. At a point where $f$ is differentiable with a nonzero gradient, the gradient is orthogonal to the level set. A level set of a function of two variables is a contour line of its graph.

Explanations

L1 human

On a map of a hill, a level set is the set of all places at one fixed height. For the bowl $f(x, y) = x^2 + y^2$, each level set at a positive height is a circle, and the circles grow as the height grows.