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Question: Show that $f(x,y)$ is homogeneous of degree zero if and only if there exists a function $g(t)$ of one variable such that $f(x,y)=g(\frac{y}{x})$

Proving the "if" part is fine, one way is:

If $f(x,y)=g(\frac{y}{x})$ then for some constant $\lambda$ we have $f(\lambda x, \lambda y)=g(\frac{\lambda y}{\lambda x})=g(\frac{y}{x})=f(x,y)={\lambda}^0f(x,y)$ and so $f(x,y)$ is homogeneous of degree zero.

But how can I prove the "only if" part? i.e. if I start with the assumption that g is anything other than of the form of a function of (y/x) that f(x,y) cannot then be homogeneous of degree zero?

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  • $\begingroup$ Try $g(y):=f(1,y).$ $\endgroup$ Commented Sep 11, 2023 at 13:20

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Write down the definition of homogeneous of degree zero, then try to choose $\lambda$ wisely.

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    $\begingroup$ Thank you both for your help. Yes, say that lambda = a then if f(x,y) is homogeneous of degree 0 then we have f(ax,ay)=f(x,y) for all a > 0. So I could choose a = 1/x for any non-zero x, then f(ax,ay) = f(1,y/x) which is just a function of (y/x). Is that sound reasoning? Feels like I've used x twice, once as a constant and once as a variable. $\endgroup$ Commented Sep 11, 2023 at 17:07
  • $\begingroup$ That is the way to go, yes! $\endgroup$ Commented Sep 11, 2023 at 18:18

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