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From: eklhad@ihnet.UUCP (K. A. Dahlke)
Newsgroups: net.math
Subject: How Many Continuous Functions Are There
Message-ID: <310@ihnet.UUCP>
Date: Fri, 4-Oct-85 10:00:08 EDT
Article-I.D.: ihnet.310
Posted: Fri Oct  4 10:00:08 1985
Date-Received: Sat, 5-Oct-85 06:42:56 EDT
Distribution: net
Organization: AT&T Bell Laboratories
Lines: 34

< What do you mean the line-eating bug is fixed?!?! >

I sort of enjoyed this problem, maybe you will too.
First, some notation for Cantor's various infinities is appropriate.
Let me call the cardinality of the integers A0 (sorry purists,
that funny symbol just isn't on my HP keyboard).
Somewhat predictably, let me call the cardinality of the reals C.
I shall call the cardinality of the set of real valued functions CC.
By the way, is there a standard symbol for this flavor of infinity?
My question is: how many *continuous* real valued functions are there?

The set is at least C, since F(x) = R (R any real number) is a valid
continuous function.  Is the set CC?

I first reviewed my diagonalization argument for *unconstrained* functions.
If the set of real valued functions is C, there is some correspondence
map between R1 and functions of R1.
Construct a new function Y(), where Y(x) = F[x](x) + 1.
In other words, to compute Y(x),
take the function associated with x, evaluate it at x, and add one.
This new function cannot be associated with any real number.
Fine for unconstrained functions, but Y() is not necessarily continuous.
Back to the drawing board.

It turns out that the set of continuous functions is C,
but I won't spoil things by giving the proof now.
This means, there is (theoretically) a procedure for ordering all
continuous functions.  Is x^2+3 > sin(x)*x?
I guess continuity is a much stronger constraint than I had realized.
There is an unimaginable sea of fuzzy jumpy functions out there.
-- 
	When You ferst realise that you're net artical contains spelling and
	grammatically errors, and sentense fragments.
			Karl Dahlke    ihnp4!ihnet!eklhad


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