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#('''{{visible anchor|Strict positive homogeneity|Strictly positive homogeneous|text=Strict}}''') '''{{visible anchor|Positive homogeneity|Positive homogeneous|Positively homogeneous}}''': <math>f(rx) = r f(x)</math> for all <math>x \in X</math> and all {{em|positive}} real <math>r > 0.</math>
#* This property is often also called '''{{visible anchor|Nonnegative homogeneity|Nonnegative homogeneous|Nonnegatively homogeneous|text=nonnegative homogeneity}}''' because for a function valued in a vector space or field, it is [[logically equivalent]] to: <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><ref group=proof>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> However, for a function valued in the [[extended real numbers]] <math>[-\infty, \infty] = \R \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 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>
#* A non-negative extended real-valued functions with this property can be characterized as being a [[Minkowski functional]].▼
#* This property is used in the definition of a [[sublinear function]].
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#'''{{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]].
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