From: israel@math.ubc.ca (Robert Israel)
Newsgroups: sci.math
Subject: Re: Calculus - continuity of real functions
Date: 26 Jan 1998 22:14:28 GMT

In article <zzpde5okxykf@forum.swarthmore.edu>, rafraf@internet-zahav.net (Dan) writes:
|> Is there a real function that is continous only at every rational
|> number 
|> 
|> (point) but not at every irrational number (point)? why ?

(Assuming the function is defined on the real numbers) the answer is no.
The reason is that the points of continuity of a function form a G_delta
set, i.e. the intersection of a countable collection of open sets.  This
can be seen by writing it as the intersection of U_m for all positive 
integers m, where x is in U_m if there is some epsilon > 0 such that
for all y and z in the interval (x-epsilon,x+epsilon), |f(y)-f(z)| < 1/m.
As a consequence of the Baire Category Theorem, the rational numbers are 
not a G_delta. 

Robert Israel                            israel@math.ubc.ca
Department of Mathematics             (604) 822-3629
University of British Columbia            fax 822-6074
Vancouver, BC, Canada V6T 1Z2