Re: stupid question?

Re: stupid question?

[Peter Freyd]

 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.

Re:stupid question?

[Michael Barr]

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.
> 

Re: Re: stupid question?

[Max Kanovitch]

 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.
 >