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        riddles >> easy >> Non-integrable integers
        
(Message started by: Benoit_Mandelbrot on May 13^th, 2004, 9:45am)

Title: Non-integrable integers
Post by Benoit_Mandelbrot on May 13^th, 2004, 9:45am

Find a function that the inputs and outputs are integers, and the function is continuous everywhere, and has no elementary anti-derivative.

Title: Re: Non-integrable integers
Post by THUDandBLUNDER on May 13^th, 2004, 10:25am

:[hide]y = ceiling[sinc{ceiling(x)}][/hide]?

Title: Re: Non-integrable integers
Post by yadayada on May 13^th, 2004, 10:50am

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..

Title: Re: Non-integrable integers
Post by towr on May 13^th, 2004, 11:22am

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)=x^x

Title: Re: Non-integrable integers
Post by Icarus on May 13^th, 2004, 4:31pm

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?

Title: Re: Non-integrable integers
Post by Benoit_Mandelbrot on May 14^th, 2004, 9:15am

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.

Title: Re: Non-integrable integers
Post by towr on May 14^th, 2004, 9:51am

meh..
x^|x| or x^(x^2) then

Title: Re: Non-integrable integers
Post by Icarus on May 14^th, 2004, 1:02pm

on 05/14/04 at 09:15:06, Benoit_Mandelbrot wrote:

and so on.

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 e^x, ln(x), trig functions, etc.)

Title: Re: Non-integrable integers
Post by SWF on May 16^th, 2004, 10:48am

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.