Dear M.M. Mawanda,
I have been asked the following question: Is it true that any function defined in a real number closed interval [a,b] (there is not a hypothesis of continuity) is bounded in an open subinterval (c,d) of [a,b]?
The real fun is about a function f such that f is unbounded in any open interval (c,d), and in addition to that: f(x+y) = f(x)+f(y).
Date: Wed, 29 Mar 2000 15:23:16 -0500 (EST) From: Peter Freyd <pjf@saul.cis.upenn.edu> Subject: categories: Re: stupid question?
M.M. Mawanda asks:
I have been asked the following question: Is it true that any function defined in a real number closed interval [a,b] (there is not a hypothesis of continuity) is bounded in an open subinterval (c,d) of [a,b]? My spontaneous was NO. Unfortunately I cannot find a counter-example to disapproved my answer. Can someone help.
No it is not true. For example, the function defined by:
f(x) = if x is irrational then 0 else if x = p/q where p and q are co-prime then q.