Give an example of degree one positively homogeneous function, (i.e. a function $f$, such that $\forall \alpha\ge0, f(\alpha x) =\alpha f(x)$) that is not linear, and $f: \mathbb{R} \to \mathbb{R}$.
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Is this a homework? Hint: What degree does the square of such a function have? – celtschk May 29 '13 at 22:48
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It is not a homework, it came up in the middle of a bigger problem. :-) – user25004 May 29 '13 at 22:52
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When you ask about degree, you assume it is a polynomial? – user25004 May 29 '13 at 22:53
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1Well, I of course meant homogeneous degree, but I was indeed thinking ultimately of a very simple polynomial. But since it's not a homework, I'll just write the answer in an answer. – celtschk May 29 '13 at 22:54
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A simple example would be $f(x)=\left|x\right|$.
About the hint in my comment: For $x\in\mathbb{R}$, $\left|x\right| = \sqrt{x^2}$.
Actually the most general function with the desired property would be $f(x)=\alpha x+\beta\left|x\right|$ with $\alpha, \beta \in\mathbb{R}$ arbitrary real constants.
celtschk
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