The note below is from Dan. It was trapped by the ever-mysterious mailing list filter. --Rich ----- Date: Sun, 1 Jun 2014 12:06:35 -0700 (GMT-07:00) Subject: Re: [math-fun] algebraic geometry query From: Dan Asimov <dasimov@earthlink.net> CORRECTION: Where I wrote "In R^n, let W be a compact connected (n-1)-dimensional submanifold with connected boundary M = bd(W)," the dimension of W should be n as below (and not n-1). Also, the affiliation is no longer current.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). -----