Support of $f \in k[x,y]/(xy,y^2)$

Question

Let $A = k[x,y]/(xy,y^2)$, where $k$ is a field, and take $f \in A$. I am working on an exercise that says prove that the support of $f$ (as a global section of the structure sheaf on Spec $A$) is either empty, the origin, or the entire space. I seem to be getting answers that is different from this, and I would appreciate if someone could show me how to do it. Thank you!

Answer 1

Show that $A_x \simeq k[x]_x$ (we're throwing out the nonreduced point $(x,y)$). This is a domain, so the support of a section there is all or nothing.