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→Euler's theorem: corrected second <math>c_+</math> to a <math>c_-</math> |
→Euler's theorem: typo |
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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>
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]].
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