Yesterday I proposed the intrinsic criterion for real 2-space zero-detection below. Nobody has yet shot it down --- which may well of course be because nobody has actually read it --- so I shall press on, making the optimistic assumption that it involves no grossly obvious errors. << Theorem: Given a continuously differentiable function f(x, y) , and a compact region R of the plane ( |R^2 ) with f nonzero on the boundary of R : f has a zero within R if and only if there is some (intrinsic critical) point (x, y) in R where *** df/dx = f = 0 ***. Proof (offered tentatively): By the implicit function theorem, the constraint f(x, y) = 0 defines a function x(y) single-valued in any interval of y where df/dx <> 0 . Such an interval meeting R is finite since R is compact, and its endpoints lie in the interior of R since f <> 0 on the boundary; therefore df/dx = 0 at those endpoints. The converse is trivial. QED.
On reflection, "region" is used rather loosely above: in fact, it seems that all that is required is that R is compact. If so, the condition cannot be relaxed in the same way that Dan did for the extrinsic version --- to f being merely C^0 continuous --- since that depended on connectivity. Which does smell just a little bit fishy ... Fred Lunnon On 6/1/14, Dan Asimov <dasimov@earthlink.net> wrote:
[I sent this 1.5 hours ago, but through some glitch it still hasn't shown up in my Inbox.]
CORRECTION:
Where I wrote "In R^n, let W be a compact connected (n-1)-dimensional submanifold with connected boundary M = bd(W)," the correct dimension of W should be n (not n-1), as corrected below.
Also, the affiliation mentioned is no longer current.
--Dan
----- Hmm, Why wouldn't C^0 (just continuous) work.
In R^n, let W be a compact connected n-dimensional submanifold with connected boundary M = bd(W).
Claim: ------ If a continuous function f: W -> R^1 (= the reals) for which 0 lies in f(W) but not in f(M), then f must have an absolute extremum in int(W) = W - M.
Proof à la Latto: ----------------- Since f is continuous on the compact set W, it must have both an absolute minimum and an absolute maximum on W. Since f is nonzero on the connected set M, f must take M into either (0,oo) or (-oo,0). If the former, f must have a local minimum in int(W); if the latter f must have a local maximum in int(W). [ ]
Corollary: ---------- If f is also differentiable on int(W), then it has a critical point there (since any local extremum x of a function differentiable in a neighborhood of x must have a critical point at x). -----