View source for Homogeneous function

{{Short description|Function with a multiplicative scaling behaviour}}
{{More footnotes|date=July 2018}}
{{for|homogeneous linear maps|Graded vector space#Homomorphisms}}
In [[mathematics]], a '''homogeneous function''' is a [[function of several variables]] such that the following holds: If each of the function's arguments is multiplied by the same [[scalar (mathematics)|scalar]], then the function's value is multiplied by some power of this scalar; the power is called the '''degree of homogeneity''', or simply the ''degree''. That is, if {{mvar|k}} is an integer, a function {{mvar|f}} of {{mvar|n}} variables is homogeneous of degree {{mvar|k}} if
:<math>f(sx_1,\ldots, sx_n)=s^k f(x_1,\ldots, x_n)</math>
for every <math>x_1, \ldots, x_n,</math> and <math>s\ne 0.</math> 

For example, a [[homogeneous polynomial]] of degree {{mvar|k}} defines a homogeneous function of degree {{mvar|k}}.

The above definition extends to functions whose [[domain of a function|domain]] and [[codomain]] are [[vector space]]s over a [[Field (mathematics)|field]] {{mvar|F}}: a function <math>f : V \to W</math> between two {{mvar|F}}-vector spaces is ''homogeneous'' of degree <math>k</math> if
{{NumBlk|:|<math>f(s \mathbf{v}) = s^k f(\mathbf{v})</math>|{{EquationRef|1}}}}
for all nonzero <math>s \in F</math> and <math>v \in V.</math> This definition is often further generalized to functions whose domain is not {{mvar|V}}, but a [[cone (linear algebra)|cone]] in {{mvar|V}}, that is, a subset {{mvar|C}} of {{mvar|V}} such that <math>\mathbf{v}\in C</math> implies <math>s \mathbf{v}\in C</math> for every nonzero scalar {{mvar|s}}.

In the case of [[functions of several real variables]] and [[real vector space]]s, a slightly more general form of homogeneity called '''positive homogeneity''' is often considered, by requiring only that the above identities hold for <math>s > 0,</math> and allowing any real number {{mvar|k}} as a degree of homogeneity. Every homogeneous real function is ''positively homogeneous''. The converse is not true, but is locally true in the sense that (for integer degrees) the two kinds of homogeneity cannot be distinguished by considering the behavior of a function near a given point.

A [[norm (mathematics)|norm]] over a real vector space is an example of a positively homogeneous function that is not homogeneous. A special case is the [[absolute value]] of real numbers. The quotient of two homogeneous polynomials of the same degree gives an example of a homogeneous function of degree zero. This example is fundamental in the definition of [[projective scheme]]s.

== Definitions ==

The concept of a homogeneous function was originally introduced for [[functions of several real variables]]. With the definition of [[vector space]]s at the end of 19th century, the concept has been naturally extended to functions between vector spaces, since a [[tuple]] of variable values can be considered as a [[coordinate vector]]. It is this more general point of view that is described in this article. 

There are two commonly used definitions. The general one works for vector spaces over arbitrary [[field (mathematics)|fields]], and is restricted to degrees of homogeneity that are [[integer]]s.

The second one supposes to work over the field of [[real number]]s, or, more generally, over an [[ordered field]]. This definition restricts to positive values the scaling factor that occurs in the definition, and is therefore called ''positive homogeneity'', the qualificative ''positive'' being often omitted when there is no risk of confusion. Positive homogeneity leads to considering more functions as  homogeneous. For example, the [[absolute value]] and all [[norm (mathematics)|norms]] are positively homogeneous functions that are not homogeneous.

The restriction of the scaling factor to real positive values allows also considering homogeneous functions whose degree of homogeneity is any real number.

=== General homogeneity ===
Let {{mvar|V}} and {{mvar|W}} be two [[vector space]]s over a [[field (mathematics)|field]] {{mvar|F}}. A [[linear cone]] in {{mvar|V}} is a subset {{mvar|C}} of {{mvar|V}} such that
<math>sx\in C</math> for all <math>x\in C</math> and all nonzero <math>s\in F.</math>

A ''homogeneous function'' {{mvar|f}} from {{mvar|V}} to {{mvar|W}} is a [[partial function]] from {{mvar|V}} to {{mvar|W}} that has a linear cone {{mvar|C}} as its [[domain of a function|domain]], and satisfies
:<math>f(sx) = s^kf(x)</math>
for some [[integer]] {{mvar|k}}, every <math>x\in C,</math> and every nonzero <math>s\in F.</math> The integer {{mvar|k}} is called the ''degree of homogeneity'', or simply the ''degree'' of {{mvar|f}}.

A typical example of a homogeneous function of degree {{mvar|k}} is the function defined by a [[homogeneous polynomial]] of degree {{mvar|k}}. The [[rational function]] defined by the quotient of two homogeneous polynomials is a homogeneous function; its degree is the difference of the degrees of the numerator and the denominator; its ''cone of definition'' is the linear cone of the points where the value of denominator is not zero.

Homogeneous functions play a fundamental role in [[projective geometry]] since any homogeneous function {{mvar|f}} from {{mvar|V}} to {{mvar|W}} defines a well-defined function between the [[projectivization]]s of {{mvar|V}} and {{mvar|W}}. The homogeneous rational functions of degree zero (those defined by the quotient of two homogeneous polynomial of the same degree) play an essential role in the [[Proj construction]] of [[projective scheme]]s.

=== Positive homogeneity ===
When working over the [[real number]]s, or more generally over an [[ordered field]], it is commonly convenient to consider ''positive homogeneity'', the definition being exactly the same as that in the preceding section, with "nonzero {{mvar|s}}" replaced by "{{math|''s'' > 0}}" in the definitions of a linear cone and a homogeneous function. 

This change allow considering (positively) homogeneous functions with any real number as their degrees, since [[exponentiation]] with a positive real base is well defined.

Even in the case of integer degrees, there are many useful functions that are positively homogeneous without being homogeneous. This is, in particular, the case of the [[absolute value]] function and [[norm (mathematics)|norms]], which are all positively homogeneous of degree {{math|1}}. They are not homogeneous since <math>|-x|=|x|\neq -|x|</math> if <math>x\neq 0.</math> This remains true in the [[complex number|complex]] case, since the field of the complex numbers <math>\C</math> and every complex vector space can be considered as real vector spaces.

[[#Euler's theorem|Euler's homogeneous function theorem]] is a characterization of positively homogeneous [[differentiable function]]s, which may be considered as the ''fundamental theorem on homogeneous functions''.

==Examples==

[[File:HomogeneousDiscontinuousFunction.gif|thumb|A homogeneous function is not necessarily [[Continuous function|continuous]], as shown by this example. This is the function <math>f</math> defined by <math>f(x,y) = x</math> if <math>xy > 0</math> and <math>f(x, y) = 0</math> if <math>xy \leq 0.</math> This function is homogeneous of degree 1, that is, <math>f(s x, s y) = s f(x,y)</math> for any real numbers <math>s, x, y.</math> It is discontinuous at <math>y = 0, x \neq 0.</math>]]

===Simple example===
The function <math>f(x, y) = x^2 + y^2</math> is homogeneous of degree 2:
<math display="block">f(tx, ty) = (tx)^2 + (ty)^2 =  t^2 \left(x^2 + y^2\right) =  t^2 f(x, y).</math>

===Absolute value and norms===
The [[absolute  value]] of a [[real number]] is a positively homogeneous function of degree {{math|1}}, which is not homogeneous, since <math>|sx|=s|x|</math> if <math>s>0,</math> and <math>|sx|=-s|x|</math>  if <math>s<0.</math>

The absolute value of a [[complex number]] is a positively homogeneous function of degree <math>1</math> over the real numbers (that is, when considering the complex numbers as a [[vector space]] over the real numbers). It is not homogeneous, over the real numbers as well as over the complex numbers.

More generally, every [[norm (mathematics)|norm]] and [[seminorm]] is a positively homogeneous function of degree {{math|1}} which is not a homogeneous function. As for the absolute value, if the norm or semi-norm is defined on a vector space over the complex numbers, this vector space has to be considered as vector space over the real number for applying the definition of a positively homogeneous function.

===Linear functions===
Any [[linear map]] <math>f : V \to W</math> between [[vector space]]s over a [[field (mathematics)|field]] {{mvar|F}} is homogeneous of degree 1, by the definition of linearity:
<math display="block">f(\alpha \mathbf{v}) = \alpha f(\mathbf{v})</math>
for all <math>\alpha \in {F}</math> and <math>v \in V.</math>

Similarly, any [[Multilinear map|multilinear function]] <math>f : V_1 \times V_2 \times \cdots V_n \to W</math> is homogeneous of degree <math>n,</math> by the definition of multilinearity:
<math display="block">f\left(\alpha \mathbf{v}_1, \ldots, \alpha \mathbf{v}_n\right) = \alpha^n f(\mathbf{v}_1, \ldots, \mathbf{v}_n)</math>
for all <math>\alpha \in {F}</math> and <math>v_1 \in V_1, v_2 \in V_2, \ldots, v_n \in V_n.</math>

===Homogeneous polynomials===
{{main article|Homogeneous polynomial}}
[[Monomials]] in <math>n</math> variables define homogeneous functions <math>f : \mathbb{F}^n \to \mathbb{F}.</math> For example,
<math display="block">f(x, y, z) = x^5 y^2 z^3 \,</math>
is homogeneous of degree 10 since
<math display="block">f(\alpha x, \alpha y, \alpha z) = (\alpha x)^5(\alpha y)^2(\alpha z)^3 = \alpha^{10} x^5 y^2 z^3 = \alpha^{10} f(x, y, z). \,</math>
The degree is the sum of the exponents on the variables; in this example, <math>10 = 5 + 2 + 3.</math>

A [[homogeneous polynomial]] is a [[polynomial]] made up of a sum of monomials of the same degree. For example,
<math display="block">x^5 + 2x^3 y^2 + 9xy^4</math>
is a homogeneous polynomial of degree 5. Homogeneous polynomials also define homogeneous functions.

Given a homogeneous polynomial of degree <math>k</math> with real coefficients that takes only positive values, one gets a positively homogeneous function of degree <math>k/d</math> by raising it to the power <math>1 / d.</math> So for example, the following function is positively homogeneous of degree 1 but not homogeneous:
<math display="block">\left(x^2 + y^2 + z^2\right)^\frac{1}{2}.</math>

===Min/max===
For every set of weights <math>w_1,\dots,w_n,</math> the following functions are positively homogeneous of degree 1, but not homogeneous:
* <math>\min\left(\frac{x_1}{w_1}, \dots, \frac{x_n}{w_n}\right)</math> ([[Leontief utilities]])
* <math>\max\left(\frac{x_1}{w_1}, \dots, \frac{x_n}{w_n}\right)</math>

===Rational functions===
[[Rational function]]s formed as the ratio of two {{em|homogeneous}} polynomials are homogeneous functions in their [[domain of a function|domain]], that is, off of the [[linear cone]] formed by the [[zero of a function|zeros]] of the denominator. Thus, if <math>f</math> is homogeneous of degree <math>m</math> and <math>g</math> is homogeneous of degree <math>n,</math> then <math>f / g</math> is homogeneous of degree <math>m - n</math> away from the zeros of <math>g.</math>

===Non-examples===
The homogeneous [[real functions]] of a single variable have the form <math>x\mapsto cx^k</math> for some constant {{mvar|c}}. So, the [[affine function]] <math>x\mapsto x+5,</math> the [[natural logarithm]] <math>x\mapsto \ln(x),</math> and the [[exponential function]] <math>x\mapsto e^x</math> are not homogeneous.

== Euler's theorem ==
Roughly speaking, '''Euler's homogeneous function theorem''' asserts that the positively homogeneous functions of a given degree are exactly the solution of a specific [[partial differential equation]]. More precisely:

{{Math theorem
| name = Euler's homogeneous function theorem
| math_statement = If {{mvar|f}} is a [[partial function|(partial) function]] of {{mvar|n}} real variables that is positively homogeneous of degree {{mvar|k}}, and [[continuously differentiable]] in some open subset of <math>\R^n,</math> then it satisfies in this open set the [[partial differential equation]]
<math display="block">k\,f(x_1, \ldots,x_n)=\sum_{i=1}^n x_i\frac{\partial f}{\partial x_i}(x_1, \ldots,x_n).</math>

Conversely, every maximal continuously differentiable solution of this partial differentiable equation is a positively homogeneous function of degree {{mvar|k}}, defined on a positive cone (here, ''maximal'' means that the solution cannot be prolongated to a function with a larger domain).
}}

{{Math proof|title=Proof|proof=  
For having simpler formulas, we set <math>\mathbf x=(x_1, \ldots, x_n).</math>
The first part results by using the [[chain rule]] for differentiating both sides of the equation <math>f(s\mathbf x ) = s^k f(\mathbf x)</math> with respect to <math>s,</math> and taking the limit of the result when {{mvar|s}} tends to {{math|1}}.

The converse is proved by integrating a simple [[differential equation]].
Let <math>\mathbf{x}</math> be in the interior of the domain of {{mvar|f}}. For {{mvar|s}} sufficiently close to {{math|1}}, the function
<math display="inline"> g(s) = f(s \mathbf{x})</math> is well defined. The partial differential equation implies that 
<math display=block>
sg'(s)= k f(s \mathbf{x})=k g(s).
</math>
The solutions of this [[linear differential equation]] have the form <math>g(s)=g(1)s^k.</math>
Therefore, <math display="block"> f(s \mathbf{x}) = g(s) = s^k g(1) = s^k f(\mathbf{x}),</math> if {{mvar|s}} is sufficiently close to {{math|1}}. If this solution of the partial differential equation would not be defined for all positive {{mvar|s}}, then the [[functional equation]] would allow to prolongate the solution, and the partial differential equation implies that this prolongation is unique. So, the domain of a maximal solution of the partial differential equation is a linear cone, and the solution is positively homogeneous of degree {{mvar|k}}. <math>\square</math>
}}

As a consequence, if <math>f : \R^n \to \R</math> is continuously differentiable and homogeneous of degree <math>k,</math> its first-order [[partial derivative]]s <math>\partial f/\partial x_i</math> are homogeneous of degree <math>k - 1.</math> 
This results from Euler's theorem by differentiating the partial differential equation with respect to one variable.

In the case of a function of a single real variable (<math>n = 1</math>), the theorem implies that a continuously differentiable and positively homogeneous function of degree {{mvar|k}} has the form <math>f(x)=c_+ x^k</math> for <math>x>0</math> and <math>f(x)=c_- x^k</math> for <math>x<0.</math> The constants <math>c_+</math> and <math>c_-</math> are not necessarily the same, as it is the case for the [[absolute value]].

==Application to differential equations==

{{main article|Homogeneous differential equation}}
The substitution <math>v = y / x</math> converts the [[ordinary differential equation]]
<math display="block">I(x, y)\frac{\mathrm{d}y}{\mathrm{d}x} + J(x,y) = 0,</math>
where <math>I</math> and <math>J</math> are homogeneous functions of the same degree, into the [[Separation of variables|separable differential equation]]
<math display="block">x \frac{\mathrm{d}v}{\mathrm{d}x} = - \frac{J(1,v)}{I(1,v)} - v.</math>

==Generalizations ==
=== Homogeneity under a monoid action ===

The definitions given above are all specialized cases of the following more general notion of homogeneity in which <math>X</math> can be any set (rather than a vector space) and the real numbers can be replaced by the more general notion of a [[monoid]]. 

Let <math>M</math> be a [[monoid]] with identity element <math>1 \in M,</math> let <math>X</math> and <math>Y</math> be sets, and suppose that on both <math>X</math> and <math>Y</math> there are defined monoid actions of <math>M.</math> Let <math>k</math> be a non-negative integer and let <math>f : X \to Y</math> be a map. Then <math>f</math> is said to be {{em|homogeneous of degree <math>k</math> over <math>M</math>}} if for every <math>x \in X</math> and <math>m \in M,</math>
<math display="block">f(mx) = m^k f(x).</math> 
If in addition there is a function <math>M \to M,</math> denoted by <math>m \mapsto |m|,</math> called an {{em|[[absolute value]]}} then <math>f</math> is said to be {{em|absolutely homogeneous of degree <math>k</math> over <math>M</math>}} if for every <math>x \in X</math> and <math>m \in M,</math>
<math display="block">f(mx) = |m|^k f(x).</math>

A function is {{em|homogeneous over <math>M</math>}} (resp. {{em|absolutely homogeneous over <math>M</math>}}) if it is homogeneous of degree <math>1</math> over <math>M</math> (resp. absolutely homogeneous of degree <math>1</math> over <math>M</math>). 

More generally, it is possible for the symbols <math>m^k</math> to be defined for <math>m \in M</math> with <math>k</math> being something other than an integer (for example, if <math>M</math> is the real numbers and <math>k</math> is a non-zero real number then <math>m^k</math> is defined even though <math>k</math> is not an integer). If this is the case then <math>f</math> will be called {{em|homogeneous of degree <math>k</math> over <math>M</math>}} if the same equality holds:
<math display="block">f(mx) = m^k f(x) \quad \text{ for every } x \in X \text{ and } m \in M.</math>

The notion of being {{em|absolutely homogeneous of degree <math>k</math> over <math>M</math>}} is generalized similarly.

===Distributions (generalized functions)===

{{main article|Homogeneous distribution}}
A continuous function <math>f</math> on <math>\R^n</math> is homogeneous of degree <math>k</math> if and only if
<math display="block">\int_{\R^n} f(tx) \varphi(x)\, dx = t^k \int_{\R^n} f(x)\varphi(x)\, dx</math>
for all [[compactly supported]] [[test function]]s <math>\varphi</math>; and nonzero real <math>t.</math> Equivalently, making a [[integration by substitution|change of variable]] <math>y = tx,</math> <math>f</math> is homogeneous of degree <math>k</math> if and only if
<math display="block">t^{-n}\int_{\R^n} f(y)\varphi\left(\frac{y}{t}\right)\, dy = t^k \int_{\R^n} f(y)\varphi(y)\, dy</math>
for all <math>t</math> and all test functions <math>\varphi.</math> The last display makes it possible to define homogeneity of [[Distribution (mathematics)|distributions]]. A distribution <math>S</math> is homogeneous of degree <math>k</math> if
<math display="block">t^{-n} \langle S, \varphi \circ \mu_t \rangle = t^k \langle S, \varphi \rangle</math>
for all nonzero real <math>t</math> and all test functions <math>\varphi.</math> Here the angle brackets denote the pairing between distributions and test functions, and <math>\mu_t : \R^n \to \R^n</math> is the mapping of scalar division by the real number <math>t.</math>

==Glossary of name variants==
{{or section|date=December 2021}}
Let <math>f : X \to Y</math> be a map between two [[vector space]]s over a field <math>\mathbb{F}</math> (usually the [[real number]]s <math>\R</math> or [[complex number]]s <math>\Complex</math>). If <math>S</math> is a set of scalars, such as <math>\Z,</math> <math>[0, \infty),</math> or <math>\Reals</math> for example, then <math>f</math> is said to be {{em|{{visible anchor|homogeneous over}} <math>S</math>}} if 
<math display=inline>f(s x) = s f(x)</math> for every <math>x \in X</math> and scalar <math>s \in S.</math> 
For instance, every [[additive map]] between vector spaces is {{em|{{visible anchor|homogeneous over the rational numbers}}}} <math>S := \Q</math> although it [[Cauchy's functional equation|might not be {{em|{{visible anchor|homogeneous over the real numbers}}}}]] <math>S := \R.</math>

The following commonly encountered special cases and variations of this definition have their own terminology:
#({{em|{{visible anchor|Strict positive homogeneity|Strictly positive homogeneous|text=Strict}}}}) {{em|{{visible anchor|Positive homogeneity|Positive homogeneous|Positively homogeneous}}}}:{{sfn|Schechter|1996|pp=313-314}} <math>f(rx) = r f(x)</math> for all <math>x \in X</math> and all {{em|positive}} real <math>r > 0.</math>
#* When the function <math>f</math> is valued in a vector space or field, then this property is [[Logical equivalence|logically equivalent]]<ref group=proof name=posHomEquivToNonnegHom /> to {{em|{{visible anchor|Nonnegative homogeneity|Nonnegative homogeneous|Nonnegatively homogeneous|text=nonnegative homogeneity}}}}, which by definition means:{{sfn|Kubrusly|2011|p=200}} <math>f(rx) = r f(x)</math> for all <math>x \in X</math> and all {{em|non-negative}} real <math>r \geq 0.</math> It is for this reason that positive homogeneity is often also called nonnegative homogeneity. However, for functions valued in the [[extended real numbers]] <math>[-\infty, \infty] = \Reals \cup \{\pm \infty\},</math> which appear in fields like [[convex analysis]], the multiplication <math>0 \cdot f(x)</math> will be undefined whenever <math>f(x) = \pm \infty</math> and so these statements are not necessarily always interchangeable.<ref group=note>However, if such an <math>f</math> satisfies <math>f(rx) = r f(x)</math> for all <math>r > 0</math> and <math>x \in X,</math> then necessarily <math>f(0) \in \{\pm \infty, 0\}</math> and whenever <math>f(0), f(x) \in \R</math> are both real then <math>f(r x) = r f(x)</math> will hold for all <math>r \geq 0.</math></ref>
#* This property is used in the definition of a [[sublinear function]].{{sfn|Schechter|1996|pp=313-314}}{{sfn|Kubrusly|2011|p=200}}
#* [[Minkowski functional]]s are exactly those non-negative extended real-valued functions with this property.
#{{em|{{visible anchor|Real homogeneity|Real homogeneous}}}}: <math>f(rx) = r f(x)</math> for all <math>x \in X</math> and all real <math>r.</math>
#* This property is used in the definition of a {{em|real}} [[linear functional]].
#{{em|{{visible anchor|Homogeneity|Homogeneous}}}}:{{sfn|Kubrusly|2011|p=55}} <math>f(sx) = s f(x)</math> for all <math>x \in X</math> and all scalars <math>s \in \mathbb{F}.</math> 
#* It is emphasized that this definition depends on the scalar field <math>\mathbb{F}</math> underlying the domain <math>X.</math>
#* This property is used in the definition of [[linear functional]]s and [[linear map]]s.{{sfn|Kubrusly|2011|p=200}}
#{{em|[[Semilinear map|{{visible anchor|Conjugate homogeneity|Conjugate homogeneous}}]]}}:{{sfn|Kubrusly|2011|p=310}} <math>f(sx) = \overline{s} f(x)</math> for all <math>x \in X</math> and all scalars <math>s \in \mathbb{F}.</math>
#* If <math>\mathbb{F} = \Complex</math> then <math>\overline{s}</math> typically denotes the [[complex conjugate]] of <math>s</math>. But more generally, as with [[semilinear map]]s for example, <math>\overline{s}</math> could be the image of <math>s</math> under some distinguished automorphism of <math>\mathbb{F}.</math> 
#* Along with [[Additive map|additivity]], this property is assumed in the definition of an [[antilinear map]]. It is also assumed that one of the two coordinates of a [[sesquilinear form]] has this property (such as the [[inner product]] of a [[Hilbert space]]).

All of the above definitions can be generalized by replacing the condition <math>f(rx) = r f(x)</math> with <math>f(rx) = |r| f(x),</math> in which case that definition is prefixed with the word {{nowrap|"{{em|absolute}}"}} or {{nowrap|"{{em|absolutely}}."}} 
For example,
<ol start=5>
<li>{{em|{{visible anchor|Absolute homogeneity|Absolute homogeneous|Absolutely homogeneous}}}}:{{sfn|Kubrusly|2011|p=200}} <math>f(sx) = |s| f(x)</math> for all <math>x \in X</math> and all scalars <math>s \in \mathbb{F}.</math>
* This property is used in the definition of a [[seminorm]] and a [[Norm (mathematics)|norm]].
</li>
</ol>

If <math>k</math> is a fixed real number then the above definitions can be further generalized by replacing the condition <math>f(rx) = r f(x)</math> with <math>f(rx) = r^k f(x)</math> (and similarly, by replacing <math>f(rx) = |r| f(x)</math> with <math>f(rx) = |r|^k f(x)</math> for conditions using the absolute value, etc.), in which case the homogeneity is said to be {{nowrap|"{{em|of degree <math>k</math>}}"}} (where in particular, all of the above definitions are {{nowrap|"{{em|of degree <math>1</math>}}"}}).
For instance, 
<ol start=6>
<li>{{em|{{visible anchor|Real homogeneity of degree}} <math>k</math>}}: <math>f(rx) = r^k f(x)</math> for all <math>x \in X</math> and all real <math>r.</math>
</li>
<li>{{em|{{visible anchor|Homogeneity of degree}} <math>k</math>}}: <math>f(sx) = s^k f(x)</math> for all <math>x \in X</math> and all scalars <math>s \in \mathbb{F}.</math>
</li>
<li>{{em|{{visible anchor|Absolute real homogeneity of degree}} <math>k</math>}}: <math>f(rx) = |r|^k f(x)</math> for all <math>x \in X</math> and all real <math>r.</math>
</li>
<li>{{em|{{visible anchor|Absolute homogeneity of degree}} <math>k</math>}}: <math>f(sx) = |s|^k f(x)</math> for all <math>x \in X</math> and all scalars <math>s \in \mathbb{F}.</math>
</li>
</ol>

A nonzero [[continuous function]] that is homogeneous of degree <math>k</math> on <math>\R^n \backslash \lbrace 0 \rbrace</math> extends continuously to <math>\R^n</math> if and only if <math>k > 0.</math>

== See also ==
* [[Homogeneous space]]
* {{annotated link|Triangle center function}}

== Notes ==
{{reflist|group=note}}

'''Proofs'''

{{reflist|group=proof|refs=
<ref group=proof name=posHomEquivToNonnegHom>Assume that <math>f</math> is strictly positively homogeneous and valued in a vector space or a field. Then <math>f(0) = f(2 \cdot 0) = 2 f(0)</math> so subtracting <math>f(0)</math> from both sides shows that <math>f(0) = 0.</math> Writing <math>r := 0,</math> then for any <math>x \in X,</math> <math>f(r x) = f(0) = 0 = 0 f(x) = r f(x),</math> which shows that <math>f</math> is nonnegative homogeneous.</ref>
}}

==References==
{{reflist}}

==Sources==
{{sfn whitelist|CITEREFKubrusly2011}}
* {{cite book|last=Blatter|first=Christian|title=Analysis II |edition=2nd |publisher=Springer Verlag|year=1979|language=German|isbn=3-540-09484-9|pages=188|chapter=20. Mehrdimensionale Differentialrechnung, Aufgaben, 1.}}
* {{Kubrusly The Elements of Operator Theory 2nd Edition 2011}} <!--{{sfn|Kubrusly|2011|p=}}-->
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!--{{sfn|Schaefer|Wolff|1999|p=}}-->
* {{Schechter Handbook of Analysis and Its Foundations}} <!--{{sfn|Schechter|1996|p=}}-->

==External links==
* {{springer|title=Homogeneous function|id=p/h047670}}
* {{MathWorld|title=Euler's Homogeneous Function Theorem|urlname=EulersHomogeneousFunctionTheorem|author=Eric Weisstein}}

[[Category:Linear algebra]]
[[Category:Differential operators]]
[[Category:Types of functions]]
[[Category:Leonhard Euler]]