Re: -- strict local mimima and level curves

On Sun, 01 Feb 2009 19:42:31 -0500, quasi <quasi@xxxxxxxx> wrote:

On Sun, 01 Feb 2009 19:33:20 -0500, quasi <quasi@xxxxxxxx> wrote:

On Sun, 01 Feb 2009 19:14:32 -0500, quasi <quasi@xxxxxxxx> wrote:

On Sun, 01 Feb 2009 15:42:58 -0800, W^3 <aderamey.addw@xxxxxxxxxxx>
wrote:

In article <ldnbo4hpjmnt1teruh5keajlm4g9hsletg@xxxxxxx>,
quasi <quasi@xxxxxxxx> wrote:

On Sun, 01 Feb 2009 11:44:35 -0500, quasi <quasi@xxxxxxxx> wrote:

Prove or disprove:

If f : R^2 --> R is continuous and if f has a strict local minimum at
(0,0) then there exists a simple closed curve S containing (0,0) such
that f is constant on S.

Of course, when I said

"a simple closed curve S containing (0,0)"

I meant

a simple closed curve S such that (0,0)
is in the region interior to S

quasi

For a > 0 let P_a denote the parabola {(x, x^2/a - a) : x in R}. As a
-> 0+, these parabolas (which are pairwise disjoint) move up, rise
faster and fold towards the nonnegative y-axis. We have U_{a>0} P_a =
R^2 minus the nonnegative y-axis. So define f : R^2 -> R by setting f
= a on P_a, f(0,y) = y for y >= 0. Then f is continuous and none of
the level sets of f is a simple closed curve.

Is f continuous on the positive y-axis?

I can see it's continuous everywhere else.

Assuming my objection is valid, I think the following minor
modification will fix it.

Define f : R^2 -> R by

f(x,y) = (a, a + |y|) for (x,y) on P_a

f(0,y) = y if y >= 0

I think it works now.

No, it doesn't work -- tthere now are simple closed curves

containing the origin

on which f is constant.

In fact, for every positive constant c, the level curve f = c is a
simple closed curve.

In trying to fix it, I made it worse -- sorry.

quasi
.

Relevant Pages

  • Re: -- strict local mimima and level curves
    ... quasi wrote: ... then there exists a simple closed curve S containing such ... Assuming my objection is valid, ... modification will fix it. ...
    (sci.math)
  • Re: -- strict local mimima and level curves
    ... quasi wrote: ... then there exists a simple closed curve S containing such ... it doesn't work -- tthere now are simple closed curves on which f ... In trying to fix it, I made it worse -- sorry. ...
    (sci.math)
  • Re: -- strict local mimima and level curves
    ... quasi wrote: ... then there exists a simple closed curve S containing such ... Is f continuous on the positive y-axis? ... Oops, no, forgot to check that. ...
    (sci.math)
  • Re: -- strict local mimima and level curves
    ... quasi wrote: ... minimum at then there exists a simple closed curve S ... of the London Mathematical Society 14, ... component of the closed set f^is equal to. ...
    (sci.math)
  • Re: -- strict local mimima and level curves
    ... quasi writes: ... then there exists a simple closed curve S containing such ... the level sets of f is a simple closed curve. ... Is f continuous on the positive y-axis? ...
    (sci.math)
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