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From: ttp@kestrel.ARPA
Newsgroups: net.math
Subject: Re: How Many Continuous Functions Are There
Message-ID: <1664@kestrel.ARPA>
Date: Tue, 8-Oct-85 14:09:38 EDT
Article-I.D.: kestrel.1664
Posted: Tue Oct  8 14:09:38 1985
Date-Received: Fri, 11-Oct-85 06:39:34 EDT
References: <310@ihnet.UUCP>, <10556@ucbvax.ARPA> <3368@pur-ee.UUCP>
Organization: Kestrel Institute, Palo Alto, CA
Lines: 12
Summary: functions on the rationals

What is the cardinality of continuous functions? (aleph 1).

try this?:

A Continuous Function on the Reals is completely defined (by
continuity) by the values it takes on the Rationals (because the
rationals are dense in the reals), thus the number of Continuous
Functions on the Reals is in One-to-One correspondence with a subset
of the Functions defined on the Rationals. The cardinality of
functions on the Rationals is the cardinality of the Reals.

-tom