What is your favorite real function?(adsbygoogle = window.adsbygoogle || []).push({});

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 q_{n}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:R^{d}->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:R^{d}->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).

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# Favorite Function

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