From: tripathi@casbah.acns.nwu.edu (Gautam Tripathi)
Newsgroups: sci.math,sci.stat.math
Subject: [Q]:opposite of lipschitzian functions ...
Date: 4 Feb 1995 07:50:56 GMT

hi all,
	i posted this article once before, but did not see it on the
newsgroups so am reposting it. if you've seen this one before
please  ignore it. hope i'm not wasting bandwidth here.
	i have the following question. does a function f() exist which 
satisfies the following property:
		|f(x,s) - f(x,t)| >= |k(x))| |s-t|     for all x
and where $x \in R^n$ $s,t \in \Theta$, $\Theta$ is some
finite dimensional subset of $R^n$ and f() is a real no.
k(x) is some function of x.
this property is exactly opposite to f() being lipschitz and i was 
wondering if such functions have been looked into. i have a feeling
that either such functions do not exist or that if they do they are 
really weird. 
	any references, hints pointers would be much appreciated ...
GT

-- 
% Gautam Tripathi                                            %
% Department of Economics        Phone (Res.):(708)-733-8137 %
% Northwestern University        Phone (Off.):(708)-491-8237 %
% Evanston, IL - 60201.          E-mail      : gt@nwu.edu    %

==============================================================================

From: rusin@washington.math.niu.edu (Dave Rusin)
Newsgroups: sci.math,sci.stat.math
Subject: Re: [Q]:opposite of lipschitzian functions ...
Date: 4 Feb 1995 09:11:29 GMT

In article <3gvbl0$oet@news.acns.nwu.edu> was written:
>	i have the following question. does a function f() exist which 
>satisfies the following property:
>		|f(x,s) - f(x,t)| >= |k(x))| |s-t|     for all x
>and where $x \in R^n$ $s,t \in \Theta$, $\Theta$ is some
>finite dimensional subset of $R^n$ and f() is a real no.
>k(x) is some function of x.

Your problem gives no indication that the values of f(x,s) with different
x's  ever interact; that is, for any individual  x  let  g(s) = f(x,s)/|k(x)|;
then your question is whether  g  has any significant properties if you know
	|g(s) - g(t) | >= | s - t |.
for all s  and  t  in  Theta. (what an unusual name for a subset of  R^n).
(If you have an x  for which  k(x) = 0, you get absolutely no information
about  f(x,s) .)

I don't know what a 'finite-dimensional' subset of  R^n  means here, but
if  Theta  is open and  g  is differentiable at a point  t  in Theta,
then for  s  near  t  we have   g(s) - g(t) = g'(t).(s-t) + E, where
E  is a quantity which diminishes faster than  s-t  does. If we apply this
to points  s  lying away from  t  in a direction perpendicular to  g'(t),
we see |g(s) - g(t)|/|s-t| will approach  0  as  s  approaches  t. This
is contrary to your hypothesis.

So, it looks to me like you've got three choices: 
	1. g  is not differentiable
	2. Theta  is not open (perhaps  Theta is a  curve in  R^n)
	3. n=1

In  R^1, the analysis above shows  |g'(t)| = lim |g(s) - g(t)| / |s-t| >=1,
and conversely if  |g'(u)| >=1  for all u  in  Theta, then (assuming
Theta is connected, i.e., an interval) by the Mean Value Theorem,
|g(s) - g(t)| = |g'(u)|.|s-t| >= |s-t|  for all s  and  t.

dave

==============================================================================

From: bruck@mtha.usc.edu (Ronald Bruck)
Newsgroups: sci.math,sci.stat.math
Subject: Re: [Q]:opposite of lipschitzian functions ...
Date: 4 Feb 1995 08:34:47 -0800

In article <3gvbl0$oet@news.acns.nwu.edu: tripathi@casbah.acns.nwu.edu (Gautam Tripathi) writes:
:hi all,
:	i posted this article once before, but did not see it on the
:newsgroups so am reposting it. if you've seen this one before
:please  ignore it. hope i'm not wasting bandwidth here.
:	i have the following question. does a function f() exist which 
:satisfies the following property:
:		|f(x,s) - f(x,t)| >= |k(x))| |s-t|     for all x
:and where $x \in R^n$ $s,t \in \Theta$, $\Theta$ is some
:finite dimensional subset of $R^n$ and f() is a real no.
:k(x) is some function of x.
:this property is exactly opposite to f() being lipschitz and i was 
:wondering if such functions have been looked into. i have a feeling
:that either such functions do not exist or that if they do they are 
:really weird. 

Oh, come on.  Your condition is that the INVERSE of f(x,.) exists and is
Lipschitzian.  Unless you're going to allow k(x) <= 0, in which case I
can think of still pithier comments.

--Ron Bruck