and for all , must be independent of , that is, given where for all , then .
So I know how to show this is true if is convex (because then I can use the Mean Value Theorem), but I am having a hard time coming up with such a function, say on a non-convex subset of , where this does not hold. Any hints or ideas would be appreciated, thanks.
Hmm, that makes sense, but my textbook (without actually showing the example) says you can find such an example if you have an open connected set in with a hole in it. But if you can cover an open connected set with overlapping convex sets, then how can such a counterexample exist?
If y>0 then regardless of whether x is positive or negative. So for all x (and f is independent of x).
If y<0 then for x<0 and also for x>0. But f(x,y) will be if x is positive, and if x is negative. So it depends on x.