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Posers and Puzzles

Posers and Puzzles

  1. Donation Acolyte
    Now With Added BA
    16 Oct '03 14:35
    Find a function on the reals which is everywhere continuous but nowhere differentiable.
  2. Standard member TheMaster37
    Kupikupopo!
    16 Oct '03 17:47
    Originally posted by Acolyte
    Find a function on the reals which is everywhere continuous but nowhere differentiable.
    Wasn't that something like...

    f(x) = 1 for x a rational number
    f(x) = 0 for x a non-rational number

    ?
  3. Standard member royalchicken
    CHAOS GHOST!!!
    16 Oct '03 18:37 / 1 edit
    Originally posted by TheMaster37
    Wasn't that something like...

    f(x) = 1 for x a rational number
    f(x) = 0 for x a non-rational number

    ?
    Your f(x) is discontinuous at every rational x.

    If r is a rational, then lim(x->r+) f(x) = 0 = lim(x->r-) f(x) < f(x) = 1.
  4. Standard member royalchicken
    CHAOS GHOST!!!
    16 Oct '03 20:01
    I'd guess it would be most useful to define the function on some interval [a,b) and than have it periodic of period b-a. Then come up with one that works on that interval.

  5. 17 Oct '03 16:01
    Originally posted by royalchicken
    I'd guess it would be most useful to define the function on some interval [a,b) and than have it periodic of period b-a. Then come up with one that works on that interval.

    sin(x)/x ?
  6. 17 Oct '03 16:04
    Originally posted by Acolyte
    Find a function on the reals which is everywhere continuous but nowhere differentiable.
    take the function f(x)= sum from n=1 to infinity (2^(-n)*cos(8^n*Pi*x))

    This example was given by Weierstrass in 1861


    Nuathala
  7. Donation Acolyte
    Now With Added BA
    17 Oct '03 18:07
    Originally posted by Nuathala
    This example was given by Weierstrass in 1861
    That's cheating!
  8. Standard member royalchicken
    CHAOS GHOST!!!
    18 Oct '03 01:17 / 1 edit
    I made one up. Let f(x) = 1+x for x in [-2,0] and 1-x if x is in [0,2] and continuing elsewhere where f(x+4) = f(x). Then let:

    F(x) = SUM (n=1 to infinity) 2^-n f(2^2^n x)

    F(x) is everywhere continuous and nowhere differentiable. In general, just find a function with regular points where the derivative fails to exist and add them up in such a way that at least one term is guaranteed to have an undefined derivative. I think many others work.

    I've got a proof; thanks for this, twas fun to work out.
  9. Standard member TheMaster37
    Kupikupopo!
    19 Oct '03 17:55
    surely there are more beautiful solutions?
  10. Standard member royalchicken
    CHAOS GHOST!!!
    19 Oct '03 21:50
    Originally posted by TheMaster37
    surely there are more beautiful solutions?
    What was the one Acolyte had in mind?
  11. 20 Oct '03 17:48
    Originally posted by royalchicken
    What was the one Acolyte had in mind?
    what was wrong with sin(x)/x ? - or can someone differentiate that? I can't!
  12. Standard member TheMaster37
    Kupikupopo!
    20 Oct '03 19:17
    Product/Quotient rule?

    (sin(x) / x)' = (sin(x))' / x + sin(x) * (1/x)' = cos(x) / x - sin(x) / x^2
  13. 21 Oct '03 23:13
    Originally posted by TheMaster37
    Product/Quotient rule?

    (sin(x) / x)' = (sin(x))' / x + sin(x) * (1/x)' = cos(x) / x - sin(x) / x^2
    doh
  14. Standard member Fiathahel
    Artist in Drawing
    23 Oct '03 17:04
    In the stochastic calculus there are functions called wiener processes. Those are almost surely nowhere differentiable. "almost surely" is because of the randomness of the function. The only problem is that they are too complicated to explain here. But they look like the graph of stockprices. The reason they are nowheree differentiable is because you can construct one taking the sum of tent functions (functions like f(x)=x for x in [0,1] f(x)=-x+1 for x in (1,2] and f(x)=0 elsewhere) Then of course you need to add incountable infinite tent functions.
    But it works
  15. 17 Nov '03 20:25
    i need to take calc.