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#76 2014-11-12 05:14:31

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
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Re: Is there such a function?

Infinite descent!


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#77 2014-11-12 05:28:42

anonimnystefy
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From: Harlan's World
Registered: 2011-05-23
Posts: 16,049

Re: Is there such a function?

Hm?


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#78 2014-11-12 05:29:17

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
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Re: Is there such a function?

How can that be non-constant?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#79 2014-11-12 06:22:53

anonimnystefy
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From: Harlan's World
Registered: 2011-05-23
Posts: 16,049

Re: Is there such a function?

It can, why do you think it can't?


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#80 2014-11-12 08:48:25

benice
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Registered: 2010-06-10
Posts: 117
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Re: Is there such a function?

anonimnystefy wrote:

Speaking of which, here is a question.

Does every non-constant periodic function have a smallest positive period?

Let f be the function defined by f(x) = 1 if x is rational and f(x) = 0 if x is irrational.
Then every rational number is a period of f.

Last edited by benice (2014-11-12 08:58:41)

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#81 2014-11-12 11:27:07

anonimnystefy
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From: Harlan's World
Registered: 2011-05-23
Posts: 16,049

Re: Is there such a function?

Yep! smile


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#82 2014-11-12 11:31:32

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
Website

Re: Is there such a function?

Great!


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#83 2014-11-12 12:01:59

ShivamS
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Registered: 2011-02-07
Posts: 3,648

Re: Is there such a function?

anonimnystefy wrote:

Does every non-constant periodic function have a smallest positive period?

Take the Dirichlet function for example.

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#84 2014-11-12 12:26:28

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,049

Re: Is there such a function?

Yep. Here's another interesting one:

Does continuity of a function f on [a,b] guarantee continuity of the function g(x)=sup{f(t)|a≤t≤x} on [a,b]? What about differentiability?


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#85 2014-11-12 12:27:15

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
Website

Re: Is there such a function?

What is sup?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#86 2014-11-12 12:30:18

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 16,049

Re: Is there such a function?

Supremum.


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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#87 2017-03-26 03:30:13

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
Website

Re: Is there such a function?

anonimnystefy wrote:

Yep. Here's another interesting one:

Does continuity of a function f on [a,b] guarantee continuity of the function g(x)=sup{f(t)|a≤t≤x} on [a,b]? What about differentiability?

Yes, this is indeed an interesting one. Continuity, I think so. Differentiablity, not sure. I will have to make a more serious attempt


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#88 2017-03-26 03:56:17

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
Website

Re: Is there such a function?

anonimnystefy wrote:

2) Is there a function with uncountably many strict extremal points?

No, there cannot be. The following is a proof, which I found on the internet, is for maximas. We could produce a similar proof for minimas and then union them to get the same result.

8M8TVacekk-uncountablemaxima.png

I cannot believe that I actually found a solution to this problem that I understood. I thoroughly remember having this conversation here on this forum two years ago, it still feels like yesterday. I remember the sequences of events that followed.

And now, here I am fully (well, at least much better) being able to comprehend this. It does turn out I have learnt something.


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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