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Suppose that $g:[0,1]^2 \rightarrow \mathbb{R}$ is a smooth function. Define the value function $$ g^*(x) = \max_{t \in [0,1]} g(x,t). $$

Question: Under what conditions would $g^*$ be a Lipschitz continuous function?

Thanks in advance for any help! Any reference would be greatly appreciated!

WaitedLeastSquare
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    Pardon me : I'd attached an earlier reference that contained mere continuity of the function. However, there exist stronger conditions under which $g^$ is differentiable. Now, if that did occur, then $g^$ would obviously be Lipschitz given domain conditions , so I'll put it out as a really minor sufficient condition : I asked about the stronger condition of differentiability here from where you can pick up references to Danskin's theorem. – Sarvesh Ravichandran Iyer Jan 17 '22 at 00:26
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    Interesting, I am exactly looking at your question (while trying to find duplicate, which I failed...) @SarveshRavichandranIyer – Arctic Char Jan 17 '22 at 00:29
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    @ArcticChar Not sure that one is an exact duplicate, although I cannot rule others. If I had to offer an argument : the difference between sufficient conditions for continuity and differentiability of the maximum are far apart. Berge's theorem assumes very little , while Danskin's theorem is quite rigid in comparison. – Sarvesh Ravichandran Iyer Jan 17 '22 at 00:32
  • @ArcticChar Yes, it does. Thanks! – WaitedLeastSquare Jan 17 '22 at 01:23
  • @SarveshRavichandranIyer Having differentiability is very helpful! Thank you! – WaitedLeastSquare Jan 17 '22 at 01:26
  • @WaitedLeastSquare Welcome, I'm happy it worked out somehow. – Sarvesh Ravichandran Iyer Jan 17 '22 at 01:29

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