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?