Level Set and Level Curve¶

Definition¶

Let $f\colon X \to \mathbb{R}$ be a real-valued function defined on a set $X\subseteq \mathbb{R}^n$ $R^{n}$ . A level set is defined as the preimage of a singleton set $\{k\}$ , i.e.,

L_k(f)=f^{-1}(\{k\}) =\{(x_1,x_2,\ldots,x_n)\in X\colon f(x_1,x_2,\ldots,x_n)=k\}.

When the number of independent variables is two, a level set is called a level curve, also known as contour line.

Interpretation¶

A level set collects all inputs at which $f$ takes one prescribed value $k$ : it answers the question "where is $f$ equal to $k$ ?". For a function of two variables these sets are usually curves in the plane, like contour lines on a map, where all points on one line have the same height.

As the preimage of $\{k\}$ , the level set $L_k(f)$ lies in the domain $X\subseteq\mathbb{R}^n$ $R^{n}$ , not on the graph. For $n=2$ it is the horizontal cut of the graph at height $z=k$ $k$ , projected onto the plane of the inputs. Despite the name, a level set need not be a curve: it is empty when $k$ is not a value of $f$ , a single point for $x^2+y^2=0$ $0$ , and all of $X$ when $f$ is constant and $k$ is its value.

A family of level curves can be read like a contour map. For equally spaced values of $k$ , level curves that lie close together mean that $f$ changes quickly there. For $f(x,y)=x^2+y^2$ the curves for $k=1,2,3,4$ are circles with radii $1,\sqrt{2},\sqrt{3},2$ , so the gaps $0.41$ , $0.32$ , $0.27$ shrink as $f$ grows faster further out.

In economics, indifference curves of a utility function and isoquants of a production function are level curves. This idea returns later in optimization with an equality constraint $g(\boldsymbol{x})=c$ $c$ : the feasible points form a level set of $g$ (method of Lagrange).

Key Results¶

Different level sets for distinct values are disjoint, and the union over all possible values covers the entire domain where the function is defined.

Examples¶

Let $f\colon \mathbb{R}^2\to\mathbb{R}$ be $f(x,y)=x^2+y^2$ $y^{2}$ . Level sets are circles centred at the origin for $k>0$ $0$ , a single point $(0,0)$ for $k=0$ $0$ , and empty for $k<0$ $0$ .

The level curves $x^2+y^2=k$ are circles of radius $\sqrt k$ for $k>0$ $0$ . They appear in red on the graph of $f$ and in blue in the $xy$ -plane.
Let $f\colon \mathbb{R}^2\to\mathbb{R}$ be $f(x,y)=x+y$ $y$ . Level sets are straight lines of slope $-1$ : $x+y=k$ $k$ . All level sets are parallel lines.

Prerequisites¶

Sources¶

Math III: Analysis · Lecture 7

Aliases: level set, level curve, contour line Symbols: element of, real numbers, square root, subset

Last updated: 27 October 2026 11:52