Homogeneous Function - Positive Homogeneity

Positive Homogeneity

In the special case of vector spaces over the real numbers, the notation of positive homogeneity often plays a more important role than homogeneity in the above sense. A function ƒ : V \ {0} → R is positive homogeneous of degree k if

for all α > 0. Here k can be any complex number. A (nonzero) continuous function homogeneous of degree k on Rn \ {0} extends continuously to Rn if and only if Re{k} > 0.

Positive homogeneous functions are characterized by Euler's homogeneous function theorem. Suppose that the function ƒ : Rn \ {0} → R is continuously differentiable. Then ƒ is positive homogeneous of degree k if and only if

This result follows at once by differentiating both sides of the equation ƒ(αy) = αkƒ(y) with respect to α, applying the chain rule, and choosing α to be 1. The converse holds by integrating.

As a consequence, suppose that ƒ : Rn → R is differentiable and homogeneous of degree k. Then its first-order partial derivatives are homogeneous of degree k − 1. The result follows from Euler's theorem by commuting the operator with the partial derivative.