Level set

Points at constant slices of x₂ = f (x₁).

Lines at constant slices of x₃ = f (x₁, x₂).

Planes at constant slices of x₄ = f (x₁, x₂, x₃).

(n − 1)-dimensional level sets for functions of the form f (x₁, x₂, …, x_n) = a₁x₁ + a₂x₂ + ⋯ + a_nx_n where a₁, a₂, …, a_n are constants, in (n + 1)-dimensional Euclidean space, for n = 1, 2, 3.

Points at constant slices of x₂ = f (x₁).

Contour curves at constant slices of x₃ = f (x₁, x₂).

Curved surfaces at constant slices of x₄ = f (x₁, x₂, x₃).

(n − 1)-dimensional level sets of non-linear functions f (x₁, x₂, …, x_n) in (n + 1)-dimensional Euclidean space, for n = 1, 2, 3.

In mathematics, a level set of a real-valued function f of n real variables is a set where the function takes on a given constant value c, that is:

${\displaystyle L_{c}(f)=\left\{(x_{1},\ldots ,x_{n})\mid f(x_{1},\ldots ,x_{n})=c\right\}~.}$

When the number of independent variables is two, a level set is called a level curve, also known as contour line or isoline; so a level curve is the set of all real-valued solutions of an equation in two variables x₁ and x₂. When n = 3, a level set is called a level surface (or isosurface); so a level surface is the set of all real-valued roots of an equation in three variables x₁, x₂ and x₃. For higher values of n, the level set is a level hypersurface, the set of all real-valued roots of an equation in n > 3 variables.

A level set is a special case of a fiber.