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Favorite Function

  1. Apr 4, 2009 #1
    What is your favorite real function?

    My top 5 are
    1. f:R->R
    [tex]f(x) = e^{-\frac{1}{x^2}}[/tex]

    for x > 0; 0 for [itex]x \le 0[/itex]. This function is infinitely differentiable everywhere, but the taylor series tells you nothing about half of the function.

    2. f:[0,1]\Q->R
    [tex]f(x) = \sum_{n=1}^\infty 2^{-n} ln |x - q_n|[/tex]

    for all rational qn in [0,1]. This function is ultra-spikey. It has a singularity at every rational number between 0 and 1, but yet the area under the curve is finite.

    3. f:Rd->R
    [tex]f(x) = (\epsilon+|x|^2)^\frac{1}{2}[/tex]

    This smooth function approximates |x|, but without the sharp point at 0.

    4. The Devil's staircase. This function is continuous, has derivative zero almost everywhere, but yet it is nonzero.

    5. f:Rd->R
    [tex]f(x) = \frac{x_1}{\sqrt{x_1^2 + x_2^2 + ... + x_k^2}}[/tex]

    for [itex]1 < k \le d[/itex]. This function is discontinuous, but the dimension of the discontinuity can be varied by adjusting k. For example if d=3 and k=3, then it is a 3D function discontinuous only at a point. If you take a derivative of it, you get a singularity (instead of a delta distribution).
  2. jcsd
  3. Apr 5, 2009 #2
    [tex]\sum_{ k = 0 }^\infty k \chi_{ [ 0, k^{ -k } ] }[/tex]. It's in [tex]L_p [ 0, 1 ][/tex] for all [tex]1 \leq p < \infty[/tex] but not in [tex]L_\infty [ 0, 1 ][/tex].
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