Level Set

The level set of a function corresponding to a real number is the set of points

${(x_1,...,x_n) in R^n:f(x_1,...,x_n)=c}.$

For example, the level set of the function corresponding to a value is the sphere with center and radius . For , the level set consists of the single point , and for it is the empty set.

For , level sets are called level curves, and for they are called level surfaces. If is continuously differentiable and its gradient is nonzero at every point of a level set that is nonempty, then that level set is locally a plane curve or surface, respectively.