# Ivory Tower Arcana 3: One for the Cult

royalchicken
Posers and Puzzles 20 Aug '04 05:07
1. royalchicken
CHAOS GHOST!!!
20 Aug '04 05:07
Prove or disprove the existence of a continuous differentiable function f:R-&gt;R that is nowhere linear such that if x is rational, f(x) is also, and if x is irrational, f(x) is also.
2. 20 Aug '04 06:22
Originally posted by royalchicken
Prove or disprove the existence of a continuous differentiable function f:R->R that is nowhere linear such that if x is rational, f(x) is also, and if x is irrational, f(x) is also.
Where are mods?!ðŸ™„
3. TheMaster37
Kupikupopo!
22 Aug '04 17:18
Suspicions to be proven:

-IF such a function exists it has to be a polynomial to get rational output with rational input.

-Polynomials aren't irrational with every irrational input (eg there is always an irrational input wich given rational output)
4. royalchicken
CHAOS GHOST!!!
22 Aug '04 17:35
Originally posted by TheMaster37
Suspicions to be proven:

-IF such a function exists it has to be a polynomial to get rational output with rational input.

Well, any ratio of two polynomials gives rational output with rational inpur (eg (x^3-9)/(x^2+3)). This doesn't necessarily give irrational output with irrationals, and the denominator may not have real roots.

-Polynomials aren't irrational with every irrational input (eg there is always an irrational input wich given rational output)

This is true. Note that irrational numbers can be associated with the degree of the lowest-degree polynomial of which they are a root. Are all irrationals roots of some polynomial?
5. TheMaster37
Kupikupopo!
22 Aug '04 18:011 edit
To the last question i know the answer immedeately. Some irrationals are no root of any FINITE polynomial, actually there are more numbers wich aren't ðŸ™‚

-IF such a function exists it has to be a polynomial to get rational output with rational input.

I change this to something that excludes nasty functions like exponentials and logarithms and such ðŸ˜€ &quot;Expressable by a polynomial or a quotient of two polinomials, and the quotient/sum of things made in that fashion, as long as it's finite&quot; lol
6. royalchicken
CHAOS GHOST!!!
22 Aug '04 18:02
Originally posted by TheMaster37
To the last question i know the answer immedeately. Some irrationals are no root of any FINITE polynomial, actually there are more numbers wich aren't ðŸ™‚
Correct. These are the transcendentals. What does this imply about our question?
7. Acolyte
23 Aug '04 16:20
Originally posted by royalchicken
Prove or disprove the existence of a continuous differentiable function f:R->R that is nowhere linear such that if x is rational, f(x) is also, and if x is irrational, f(x) is also.

f(x) = 1/(x-2) + 2 for x&lt;1; 1/x otherwise
8. royalchicken
CHAOS GHOST!!!
23 Aug '04 17:07
Originally posted by Acolyte

f(x) = 1/(x-2) + 2 for x<1; 1/x otherwise
That works ðŸ™‚.
9. 23 Aug '04 21:53
EGWIDG BLUMINCRAFT!!!

MAY ALLL THE GODS FOREVER DAMN MY LACK OF CALCULUS!!!!!
10. royalchicken
CHAOS GHOST!!!
24 Aug '04 04:44