Jeff Freeman
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 Jul 8 accepted Conditions for functions to be independent of one of their variables Jul 8 comment Conditions for functions to be independent of one of their variables Oops. Well, it's now clear to me that I was mistaken about being able to extend the function, and that continuity is necessary for $x<0 \text{ and } y=0$ so that the derivative can exist for all of $A$. Thanks for your help. If you care to sum up the discussion in an answer, I'll accept it. Jul 7 comment Conditions for functions to be independent of one of their variables Best I can figure, is that it has to do with implicit continuity. That is, if we define a jump function, with a constant for $y\lt 0$ and another constant for $y\ge 0$, then the partial derivative is $0$, but $f(x,-1)\ne f(x,1)$. Is that about right? Jul 7 asked Conditions for functions to be independent of one of their variables Apr 13 awarded Supporter Jun 18 awarded Scholar Jun 18 accepted Upper bound on number of starting positions of a grid coloring game Jun 18 comment Upper bound on number of starting positions of a grid coloring game Noticing that the perimeter of the true squares remained invariant was clever. Thank you for your insight! Jun 18 asked Upper bound on number of starting positions of a grid coloring game Feb 25 awarded Autobiographer