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.
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.
How about the following? f(x) = 0 if x is irrational and f(a/b) = a, where a/b is a fraction in lowest terms Certainly within any open interval, there are rationals of arbitrarily large numerator. For a function that is not bounded above or below, how about: f(x) = 0 if x is irrational f(a/b) = a if b is even f(a/b) = -a if b is odd -----Original Message----- From: M.M. Mawanda [mailto:mm.mawanda@nul.ls] Sent: March 29,2000 4:08 PM To: cat-dist@mta.ca Subject: categories: stupid question? 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.
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.
On Wed, 29 Mar 2000, Max Kanovitch wrote:
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).
Using AC/Zorn's Lemma, we can construct 2^(2^Aleph_0) such functions, as follows. Let B be a basis for the reals R as a rational vector space. Clearly, |B| = 2^Aleph_0. For any non-empty proper subset C of B, let g be its characteristic function (g(x)=1 if x in C, g(x)=0 otherwise) and let f be the unique linear extension of g to R. Then f is linear, and its graph is dense in R^2, since, if c and d are such that g(c)=1 and g(d)=0, then f(qc+rd) = q for all rationals q,r, and r can be varied to make qc+rd as close as desired to any given real. -- Todd Wilson Computer Science Department California State University, Fresno
It is not even true for additive functions. Take a Hamel base and send every element of the base to 1. On Wed, 29 Mar 2000, Peter Freyd wrote:
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.
participants (6)
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M.M. Mawanda -
maxkanov＠math.upenn.edu -
Michael Barr -
Peter Freyd -
Todd Wilson -
Wendt, Michael - SSMD/DMES