# Non Continuous Walrasian Demand Function

I have a silly question.

I’m trying to solve some exercises that have to do with the walrasian demand function $$x(p,w)$$ and excess demand function $$z$$.

More especifically, I’m asked to show that there is not a price vector ($$p$$ belongs to the simplex) such that $$z(p)=0$$ when $$z$$ is not continuous for all goods. As $$z$$, for a given good $$i$$, is the sum of the walrasian demand and a given endowment, I think that the previous statement is the same as saying that if $$x(p,w)$$ is not continuous then there is not $$p$$ s.t $$z(p)=0$$.

Any hint about why, if $$z$$ or $$x$$ aren’t continuous then $$p$$ doesn’t exist?

• There seems to be something missing, "$\ldots$such that $z(p)$" isn't really a statement about anything. Feb 6, 2023 at 21:29
• @Martin what is that excess demand function $z$ you talk about? I just started Advanced Microeconomics. Feb 7, 2023 at 0:49
• Sorry for the delay Nicolas. The excess demand function, as defined by Mas Colell on chapter 17, is the good’s walrasian demand function (x)minus the endowment (w). Feb 9, 2023 at 9:00
• Michael, I corrected the statement. Thanks! Feb 9, 2023 at 9:02
• It is not true that there can be no $p$ such that $z(p)=0$ when $z$ or (equivalently) $x$ is discontinuous. There may not be, but continuity is not a necessary condition. Feb 9, 2023 at 10:15