# How to show that a function is homogeneous of degree zero if and only if …

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?

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

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

• 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. Commented Sep 11, 2023 at 17:07
• That is the way to go, yes! Commented Sep 11, 2023 at 18:18