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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
    Commented Oct 19, 2019 at 23:35
  • $\begingroup$ Sorry, this was a typo. $\endgroup$
    – Aqqqq
    Commented Oct 20, 2019 at 5:14
  • $\begingroup$ Look at the Theorem of the maximum: en.wikipedia.org/wiki/Maximum_theorem $\endgroup$
    – user24622
    Commented Oct 20, 2019 at 15:49
  • $\begingroup$ @user24622 The Theorem of the Maximum will give you upper hemicontinuity of the Walrasian demand correspondence, but AFAICT this does not imply continuity (in the usual sense) even if Walrasian demand is single-valued. $\endgroup$
    – chsk
    Commented Jan 25 at 8:20

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