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Dragonfall
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 Posted: Mon Sep 29, 2008 4:26 pm Post subject: Pop Quiz |
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Prove or disprove:
There exists a function f:R->R such that:
(1) f is continuous
(2) f is not constant on any interval
(3) f has uncountably many zeros.
If you give up, I'll give you the answer. |
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RyanDzundza
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| Posted: Mon Sep 29, 2008 4:30 pm Post subject: |
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i havent got a clue what your on about so i give up, i feel stupid now 
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Dragonfall
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| Posted: Mon Sep 29, 2008 4:46 pm Post subject: |
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| Sorry. |
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baddogma
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abstract atheist
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| Posted: Mon Sep 29, 2008 7:01 pm Post subject: Re: Pop Quiz |
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| Dragonfall wrote: | Prove or disprove:
There exists a function f:R->R such that:
(1) f is continuous
(2) f is not constant on any interval
(3) f has uncountably many zeros.
If you give up, I'll give you the answer. |
God damn it Dragon, why did you have to include uncountable? Let's see, sine and cosine are continuous, not constant, and have countably infinite zeros. so take the limit of sin (kx) as k goes to infinity, and perhaps this new function will have uncountable number of zeros.
f(x)=limit as k goes to infinity of sin (kx)
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The world holds two classes of men - intelligent men without religion, and religious men without intelligence. - Abu’l‐Ala al Ma’arri
Give a man a fish and he will eat for a day; teach a man to fish and he will eat for a lifetime; give a man religion and he will die praying for a fish.
Father: Son, I don't want you hanging around Steve any more, I think he might be gay.
Son: He better be gay, he's my boyfriend. |
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Dragonfall
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| Posted: Mon Sep 29, 2008 7:50 pm Post subject: |
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Heh heh. Well without uncountable it's too easy.
But I'm afraid the function you gave is not a function at all. Well not a real -> real function anyway. |
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abstract atheist
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| Posted: Mon Sep 29, 2008 10:29 pm Post subject: |
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| Dragonfall wrote: | Heh heh. Well without uncountable it's too easy.
But I'm afraid the function you gave is not a function at all. Well not a real -> real function anyway. |
I have a proof that no such function exists.
1. All continuous functions have at most a countable number of wiggles.
2. each wiggle can intersect the x-axis at most twice.
3. Hence a continuous real valued function that's never flat can only have a countable number of wiggles.
I need to prove assumption 1, the rest follows obviously.
We will do proof by contradiction.
1. Assume a function exists with an uncountable number of wiggles.
2. That means that there must exist an interval on the x-axis, that, no matter how small I zoom in, the function has an uncountable number of wiggles in that interval.
3. The only way that can happen is if the distance between wiggles goes to zero.
4. Something bad happens.
5. Therefore, we cannot have a function with an uncountable number of wiggles.
_________________
The world holds two classes of men - intelligent men without religion, and religious men without intelligence. - Abu’l‐Ala al Ma’arri
Give a man a fish and he will eat for a day; teach a man to fish and he will eat for a lifetime; give a man religion and he will die praying for a fish.
Father: Son, I don't want you hanging around Steve any more, I think he might be gay.
Son: He better be gay, he's my boyfriend. |
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Dragonfall
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Posts: 53
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| Posted: Mon Sep 29, 2008 11:36 pm Post subject: |
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| abstract atheist wrote: |
4. Something bad happens. |
Unfortunately nothing bad happens! Think about it, sin(1/x) has "wiggles" whose pairwise distance goes to 0 around the origin.
Also, it's because a function with those properties that I asked for actually exists!
It's not some really complicated function, either. |
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abstract atheist
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| Posted: Mon Sep 29, 2008 11:54 pm Post subject: |
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| Dragonfall wrote: | | abstract atheist wrote: |
4. Something bad happens. |
Unfortunately nothing bad happens! Think about it, sin(1/x) has "wiggles" whose pairwise distance goes to 0 around the origin.
Also, it's because a function with those properties that I asked for actually exists!
It's not some really complicated function, either. |
I will find that function and report it tommorrow. Unless it's xsin(1/x), redefined to be 0 at the origin. That function is continuous, real valued but I don't know how many wiggles it has.
_________________
The world holds two classes of men - intelligent men without religion, and religious men without intelligence. - Abu’l‐Ala al Ma’arri
Give a man a fish and he will eat for a day; teach a man to fish and he will eat for a lifetime; give a man religion and he will die praying for a fish.
Father: Son, I don't want you hanging around Steve any more, I think he might be gay.
Son: He better be gay, he's my boyfriend. |
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Dragonfall
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Joined: 08 Apr 2007
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| Posted: Tue Sep 30, 2008 8:13 am Post subject: |
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| xsin(1/x) has only countably many zeros. |
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abstract atheist
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| Posted: Tue Sep 30, 2008 12:27 pm Post subject: |
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DRAGONFALL IS A FRAUD
Oh, and i don't know if I believe your friends at physics forums.
P.S. don't use the same screen name on other forums because it makes it easy for me to stalk you.
_________________
The world holds two classes of men - intelligent men without religion, and religious men without intelligence. - Abu’l‐Ala al Ma’arri
Give a man a fish and he will eat for a day; teach a man to fish and he will eat for a lifetime; give a man religion and he will die praying for a fish.
Father: Son, I don't want you hanging around Steve any more, I think he might be gay.
Son: He better be gay, he's my boyfriend. |
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Moloth
Fateless
Joined: 27 Aug 2003
Posts: 23104
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Location: Warner Robins, GA
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| Posted: Tue Sep 30, 2008 1:35 pm Post subject: |
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..when math nerds attack?
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-=The Believer is Happy; the Skeptic is Wise=-
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Last edited by Moloth on Tue Feb 30, 2026 13:61 am; edited 426 times in total |
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abstract atheist
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Posts: 598
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| Posted: Tue Sep 30, 2008 3:08 pm Post subject: |
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| Moloth wrote: | | ..when math nerds attack? |
They will, and when they do it will be way, way worse than that other take over the world scenario that I created. And by the way, Dragonfall's "function" that is real valued, continuous, and never flat on an interval (with an uncountable zeros) is really
f(x)=0 on the cantor set
f(x)=triangle inbetween cantor points.
But don't let him fool you, he didn't come up with it himself.
Of course I'm just mad because i didn't think of it.
_________________
The world holds two classes of men - intelligent men without religion, and religious men without intelligence. - Abu’l‐Ala al Ma’arri
Give a man a fish and he will eat for a day; teach a man to fish and he will eat for a lifetime; give a man religion and he will die praying for a fish.
Father: Son, I don't want you hanging around Steve any more, I think he might be gay.
Son: He better be gay, he's my boyfriend. |
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Moloth
Fateless
Joined: 27 Aug 2003
Posts: 23104
Local time: 12:50 PM
Location: Warner Robins, GA
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| Posted: Tue Sep 30, 2008 3:09 pm Post subject: |
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well, YEAH.. pshh.. i mean, how could one NOT come up with that?
>.>
_________________
-=The Believer is Happy; the Skeptic is Wise=-
www.Moloth.com
Last edited by Moloth on Tue Feb 30, 2026 13:61 am; edited 426 times in total |
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Dragonfall
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Joined: 08 Apr 2007
Posts: 53
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| Posted: Tue Sep 30, 2008 4:01 pm Post subject: |
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| f is the distance function from x to the cantor set. |
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