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Topic: Non-integrable integers (Read 517 times) |
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Benoit_Mandelbrot
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Non-integrable integers
« on: May 13th, 2004, 9:45am » |
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Find a function that the inputs and outputs are integers, and the function is continuous everywhere, and has no elementary anti-derivative.
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| « Last Edit: May 13th, 2004, 10:06am by Benoit_Mandelbrot » |
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Because of modulo, different bases, and significant digits, all numbers equal each other!
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ThudnBlunder
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Re: Non-integrable integers
« Reply #1 on: May 13th, 2004, 10:25am » |
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:y = ceiling[sinc{ceiling(x)}]?
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yadayada
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Re: Non-integrable integers
« Reply #2 on: May 13th, 2004, 10:50am » |
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Looks like it is not possible to have a function f:R -> Z which is continuous (except for the trivial constant function)
Say f is not constant. then f takes on two different values M and N. By mean value theorem, f must take on every value between M and N. Which is not possible...
Similary, there is no non-trivial continuous function g:R->Q where Q is the set of rationals..
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towr
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Re: Non-integrable integers
« Reply #3 on: May 13th, 2004, 11:22am » |
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I don't think we're looking for an f:[bbr][to][bbz], but an f:[bbr][to][bbr] with the additional properties that [forall]n[in][bbz]: f(n)[in][bbz]
so maybe, f(x)=xx
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Wikipedia, Google, Mathworld, Integer sequence DB
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Icarus
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Re: Non-integrable integers
« Reply #4 on: May 13th, 2004, 4:31pm » |
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I have two questions for Benoit.
Is towr right in his interpretation? Quite frankly, the original statement doesn't make sense.
What do you mean by "elementary" anti-derivative?
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Benoit_Mandelbrot
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Re: Non-integrable integers
« Reply #5 on: May 14th, 2004, 9:15am » |
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Well, x^x has no elementary anti-derivative, but (-2)^(-2) is a fraction, not an integer , being -1/4. Elementary functions would be a^x, logarithms, trig functions, and so on. An elementary anti-derivative should contain one or more of these. This means that f(I1)=I2, where I is any integer. Thud and Blunder's wouldn't work, because it isn't continuous. x^x won't work because any x<0 would give fractions for integers. Sorry if I didn't clarify this enough.
This function must be continuous. This function must return integers when you put in an integer. This functions must have no anti-derivative in which is made up of only elementary functions.
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| « Last Edit: May 14th, 2004, 9:21am by Benoit_Mandelbrot » |
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Icarus
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Re: Non-integrable integers
« Reply #7 on: May 14th, 2004, 1:02pm » |
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on May 14th, 2004, 9:15am, Benoit_Mandelbrot wrote:
And what is included in "and so on"? Bessel functions? Legendre functions? Elliptic integrals? logarithmic integrals? Hypergeometrics?
Or are you restricting yourself to functions constructible in finitely many steps from the operations of addition, subtraction, multiplication, division, exponentiation, and inversion?
(This would pick up ex, ln(x), trig functions, etc.)
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I wonder: Which is larger
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SWF
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Re: Non-integrable integers
« Reply #8 on: May 16th, 2004, 10:48am » |
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Another solution is to use the gamma function with argument x2+1:
When x is an integer, [Gamma](x2+1) equals the integer (x2)!.
For many continuous functions f(x), f(x)*sin([pi]*x) meets the conditions of the question, again depending on what counts as elementary.
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