Homogeneity holds but additivity does not. I'm stuck

[steelphantom]

Homework Statement

Give an example of a function f:R^2 -> R such that f(av) = a(f(v)) for all a in R and all v in R^2 but f is not linear.

Homework Equations

f(v + w) = f(v) + f(w) (Additivity)

The Attempt at a Solution

I really can't think of a function that will satisfy these properties. I know that for this to work, homogeneity must hold but additivity must not. I tried several functions, but none worked. Any hints? Thanks.

 

[Dick]

How about f((x,y))=x^(1/3)*y^(2/3)?

 

[steelphantom]

How about f((x,y))=x^(1/3)*y^(2/3)?

Thanks! I see how you came up with that now. I tried f((x, y)) = x^2 + y^2 but the first property didn't hold since a changed. Your function fixes that issue. Thanks again for the help.