Let _f_ be a function from R into itself with this property: whenever a real number _x_ is the limit of a sequence (q_n)_n of rational numbers, then lim_n f(q_n) = f(x). Does it follow that _f_ is continuous?
> Let _f_ be a function from R into itself with this > property: whenever a > real number _x_ is the limit of a sequence (q_n)_n of > rational numbers, > then lim_n f(q_n) = f(x). Does it follow that _f_ is > continuous?
>Let _f_ be a function from R into itself with this property: whenever a >real number _x_ is the limit of a sequence (q_n)_n of rational numbers, >then lim_n f(q_n) = f(x). Does it follow that _f_ is continuous?
Yes. This is actually easy.
Say f is discontinuous at 0. Let's say f(0) = 0 but there is a sequence of reals r_n -> 0 with f(r_n) > 1 for all n. For each n, consider a sequence of rationals approaching r_n, and you see that there exists a rational q_n with f(q_n) > 1/2 and |r_m - q_n| < |r_n|/2. Hence q_n -> 0, contradicting the hypothesis.
>Best regards,
>Jose Carlos Santos
David C. Ullrich
"Understanding Godel isn't about following his formal proof. That would make a mockery of everything Godel was up to." (John Jones, "My talk about Godel to the post-grads." in sci.logic.)
>> Let _f_ be a function from R into itself with this >> property: whenever a >> real number _x_ is the limit of a sequence (q_n)_n of >> rational numbers, >> then lim_n f(q_n) = f(x). Does it follow that _f_ is >> continuous?
>> Best regards,
>> Jose Carlos Santos
>no.
Giggle. How do you prove that?
(giggle. I suppose you figured you had a 50-50 chance of guessing right. Too bad about that.)
David C. Ullrich
"Understanding Godel isn't about following his formal proof. That would make a mockery of everything Godel was up to." (John Jones, "My talk about Godel to the post-grads." in sci.logic.)
>> Let _f_ be a function from R into itself with this property: whenever a >> real number _x_ is the limit of a sequence (q_n)_n of rational numbers, >> then lim_n f(q_n) = f(x). Does it follow that _f_ is continuous?
> Yes. This is actually easy.
I bet you only wrote that to make me feel bad. :-)
> Say f is discontinuous at 0. Let's say f(0) = 0 but there > is a sequence of reals r_n -> 0 with f(r_n) > 1 for all n. > For each n, consider a sequence of rationals approaching > r_n, and you see that there exists a rational q_n with > f(q_n) > 1/2 and |r_m - q_n| < |r_n|/2.
In article <7ltfs1F3fnia...@mid.individual.net>, José Carlos Santos <jcsan...@fc.up.pt> wrote:
> On 10-11-2009 15:20, David C. Ullrich wrote:
> >> Let _f_ be a function from R into itself with this property: whenever a > >> real number _x_ is the limit of a sequence (q_n)_n of rational numbers, > >> then lim_n f(q_n) = f(x). Does it follow that _f_ is continuous?
> > Yes. This is actually easy.
> I bet you only wrote that to make me feel bad. :-)
Sorry. If it makes you feel any better, although it _is_ easy it took me a minute to decide.
(Ok, I actually wrote that to make someone else feel bad, I'll let you guess who.)
> > Say f is discontinuous at 0. Let's say f(0) = 0 but there > > is a sequence of reals r_n -> 0 with f(r_n) > 1 for all n. > > For each n, consider a sequence of rationals approaching > > r_n, and you see that there exists a rational q_n with > > f(q_n) > 1/2 and |r_m - q_n| < |r_n|/2.
On 10 marras, 13:19, José Carlos Santos <jcsan...@fc.up.pt> wrote:
> Hi all,
> Let _f_ be a function from R into itself with this property: whenever a > real number _x_ is the limit of a sequence (q_n)_n of rational numbers, > then lim_n f(q_n) = f(x). Does it follow that _f_ is continuous?
Follows directly from that every real number is always a (equivalence class of) limit(s) of a cauchy sequence of rationals. You didn`t remember this?
On 10 marras, 19:48, Gc <gcut...@hotmail.com> wrote:
> On 10 marras, 13:19, José Carlos Santos <jcsan...@fc.up.pt> wrote:
> > Hi all,
> > Let _f_ be a function from R into itself with this property: whenever a > > real number _x_ is the limit of a sequence (q_n)_n of rational numbers, > > then lim_n f(q_n) = f(x). Does it follow that _f_ is continuous?
> Follows directly from that every real number is always a (equivalence > class of) limit(s) of a cauchy sequence of rationals. You didn`t > remember this?
I mean "quite" directly. I was thinking a limit changing procedure, to which, now I think, is little more complicated because I think you need to prove a equicontinuity. But it is sometimes nice to have also a direct proof.
On Tue, 10 Nov 2009 09:48:03 -0800 (PST), Gc <gcut...@hotmail.com> wrote:
>On 10 marras, 13:19, José Carlos Santos <jcsan...@fc.up.pt> wrote: >> Hi all,
>> Let _f_ be a function from R into itself with this property: whenever a >> real number _x_ is the limit of a sequence (q_n)_n of rational numbers, >> then lim_n f(q_n) = f(x). Does it follow that _f_ is continuous?
>Follows directly from that every real number is always a (equivalence >class of) limit(s) of a cauchy sequence of rationals. You didn`t >remember this?
Remmeber what, exactly? Saying every real number is an equivalence class of limits of a cauchy sequence of rationals is nonsense. A cauchy sequence has one limit, and a real _is_ the limit of a cauchy sequence, not an equivalence class of such limits.
Every real _is_ the limit of a cauchy sequence of rationals. (And one construction of the reals says a real is an equivalence class of cauchy sequences of rationals.)
David C. Ullrich
"Understanding Godel isn't about following his formal proof. That would make a mockery of everything Godel was up to." (John Jones, "My talk about Godel to the post-grads." in sci.logic.)
> On Tue, 10 Nov 2009 09:48:03 -0800 (PST),Gc<gcut...@hotmail.com> > wrote:
> >On 10 marras, 13:19, José Carlos Santos <jcsan...@fc.up.pt> wrote: > >> Hi all,
> >> Let _f_ be a function from R into itself with this property: whenever a > >> real number _x_ is the limit of a sequence (q_n)_n of rational numbers, > >> then lim_n f(q_n) = f(x). Does it follow that _f_ is continuous?
> >Follows directly from that every real number is always a (equivalence > >class of) limit(s) of a cauchy sequence of rationals. You didn`t > >remember this?
> Remmeber what, exactly? Saying every real number is an equivalence > class of limits of a cauchy sequence of rationals is nonsense. A > cauchy sequence has one limit, and a real _is_ the limit of a > cauchy sequence, not an equivalence class of such limits.
This is what I meant...
> Every real _is_ the limit of a cauchy sequence of rationals. > (And one construction of the reals says a real is an equivalence > class of cauchy sequences of rationals.)
...and somewhere when writing about it confused with this.
> On Tue, 10 Nov 2009 09:48:03 -0800 (PST), Gc <gcut...@hotmail.com> > wrote:
> >On 10 marras, 13:19, José Carlos Santos <jcsan...@fc.up.pt> wrote: > >> Hi all,
> >> Let _f_ be a function from R into itself with this property: whenever a > >> real number _x_ is the limit of a sequence (q_n)_n of rational numbers, > >> then lim_n f(q_n) = f(x). Does it follow that _f_ is continuous?
> >Follows directly from that every real number is always a (equivalence > >class of) limit(s) of a cauchy sequence of rationals. You didn`t > >remember this?
> Remmeber what, exactly? Saying every real number is an equivalence > class of limits of a cauchy sequence of rationals is nonsense. A > cauchy sequence has one limit, and a real _is_ the limit of a > cauchy sequence, not an equivalence class of such limits.
> Every real _is_ the limit of a cauchy sequence of rationals. > (And one construction of the reals says a real is an equivalence > class of cauchy sequences of rationals.)
> David C. Ullrich
> "Understanding Godel isn't about following his formal proof. > That would make a mockery of everything Godel was up to." > (John Jones, "My talk about Godel to the post-grads." > in sci.logic.)
