If the utility function $u$ is continuous and satisfies local nonsatiation, and walrasian demand function $x(p, w)$ is a function (i.e. always map to only single values), how to prove that $x(p,w)$ is continuous in $p$ (price vector) and $w$ (wealth)?
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$\begingroup$ What are $x(p,w)$ and $w(p,w)$? Are they both the same walrasian demand function? $\endgroup$ – GabMac Oct 19 '19 at 23:35
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$\begingroup$ Sorry, this was a typo. $\endgroup$ – Aqqqq Oct 20 '19 at 5:14
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$\begingroup$ Look at the Theorem of the maximum: en.wikipedia.org/wiki/Maximum_theorem $\endgroup$ – user24622 Oct 20 '19 at 15:49
