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From: cjh@petsd.UUCP (Chris Henrich)
Newsgroups: net.physics
Subject: Re: White Holes?
Message-ID: <617@petsd.UUCP>
Date: Fri, 16-Aug-85 19:43:50 EDT
Article-I.D.: petsd.617
Posted: Fri Aug 16 19:43:50 1985
Date-Received: Tue, 20-Aug-85 04:37:21 EDT
References: <3656@decwrl.UUCP>
Reply-To: cjh@petsd.UUCP (PUT YOUR NAME HERE)
Organization: Perkin-Elmer DSG, Tinton Falls, N.J.
Lines: 70
Summary: What do yuo mean by "continuous function"?

[]
In article <3656@decwrl.UUCP> williams@kirk.DEC (John Williams 223-3402) writes:
>	My intuition tells me that the universe is a continuous function.
>This would not necessitate a creator, where would a creator come from?
>A mathematical function simply exists.

As I remember the definition of "continuous function" from
various mathematical courses and books, it is this:

In an elementary context:

	f is continuous at x0 if the limit of f(x) as x 
	approaches x0 exists, and is equal to f(x0). 
	(Perfunctory apologies for a crude approximation
	to mathematical notation.)  The function is
	continuous if it is "continuous at x0" for every
	x0 in its domain.

Another definition, roughly equivalent to the first, is this:

	The function f is continuous at x0 if, for every
	Eps > 0, there exists a Del > 0 such that

	   | x - x0 | < Del  ==>  |f(x) - f(x0)| < Eps.

Or, if one has the notion of a "distance" or "metric" on the
domain and another on the range, then

	The function f is continuous at x0 if, for every
	Eps > 0, there exists a Del > 0 such that

	dist(x, x0) < Del  ==>  dist( f(x), f(x0) ) < Eps.

The most general definition known to me is like this.

	Let X and Y be topological spaces. (I.e. they are
	sets, and for each a class of "open" subsets is
	determined somehow.)  Let f be a function whose domain
	is X and whose range is contained in Y.
	Then f is continuous if, for every open subset S of
	Y, the inverse image of S under f is an open subset 
	of X.

References are abundant.  Books on "mathematical analysis"
or "functional analysis" usually explain some or all of these
definitions.  (Good ones include _Analysis_I_ and
_Analysis_II_ by S. Lang.)  See also introductory books on
"topology."  There are good ones by Kelley and by Dugundji,
and many more.

If we try to apply these definitions to Mr. Williams's
statements, nothing much happens.  To call the universe a
continunous function seems to me to be saying remarkably
little.  What, pray tell, is the domain, and what is the
range?  These questions being settled, can you tell us
something about the values the function takes at various
elements of the range?

Or is there some completely different idea that is going by
the name "continuous function"?  If so, what is it?


Regards,
Chris

--
Full-Name:  Christopher J. Henrich
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