Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.1 6/24/83; site petsd.UUCP Path: utzoo!watmath!clyde!burl!ulysses!mhuxr!mhuxt!houxm!vax135!petsd!cjh 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 UUCP: ..!(cornell | ariel | ukc | houxz)!vax135!petsd!cjh US Mail: MS 313; Perkin-Elmer; 106 Apple St; Tinton Falls, NJ 07724 Phone: (201) 758-7288