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From: janw@inmet.UUCP
Newsgroups: net.math
Subject: Re: How Many Continuous Functions Are Th
Message-ID: <5700012@inmet.UUCP>
Date: Mon, 7-Oct-85 20:41:00 EDT
Article-I.D.: inmet.5700012
Posted: Mon Oct  7 20:41:00 1985
Date-Received: Sat, 12-Oct-85 16:41:32 EDT
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Nf-From: inmet!janw    Oct  7 20:41:00 1985


A continuous function is uniquely defined by its values over rational
arguments. Rational numbers are countable; thus a continuous real
function is uniquely defined by a sequence of real numbers.
To answer the problem, it is enough to assign every sequence
of real numbers a unique real number.
We can assume all the numbers to be in [0,1).
Then a sequence S of them can be represented by an infinite matrix
of digits such that s[i,j] is the digit in the i-th position
in the j-th member of the sequence. Traversing this matrix in the
usual diagonal fashion : concatenating the finite diagonals
in the order of increasing length (s[1,1]s[2,1]s[1,2]s[3,1] ...),
we obtain a digit sequence uniquely defined for the matrix;  pre-
ceded by "0.", this yields the real number corresponding to S.

		Jan Wasilewsky