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I need help understanding this problem:

$$f(x,y)=\begin{cases} \dfrac{y^4}{x}, & x>2, y>0 \\[6pt] y^3,& (x,y)\in E \end{cases}$$

$$D:=R^2 \smallsetminus \{{(x,y)\in R^2 : x=2, y\ge 0}\}$$

$$E:=D\smallsetminus \{{(x,y)\in R^2 : x>2, y> 0}\}$$

This function is locally homogeneous in $D$, but it's not homogeneous in its domain. I know if $f(tx,ty)=t^nf(x,y)$ function is homogeneous with degree of $n$. And for local homogenity I need to use Euler's theorem for homogeneous function. And every function that is localy homogeneous is homogeneous on convex domain. And in this case function is homogeneous with degree of $3$. But I don't understand why is it not homogeneous on its domain?

Michael Hardy
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Lamija37
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  • Note proper use of \smallsetminus in MathJax, as in my edit to the question. You can also use \setminus and the result looks a bit different: $A\smallsetminus B$ versus $A\setminus B.$ But I wonder why you write $ \text{\text{$x,y\in E$}} $ within MathJax instead of just x,y\in E. $\qquad$ – Michael Hardy Jul 16 '17 at 16:41

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