Favorite Real Function: Top 5 Picks

[maze]

What is your favorite real function?

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

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


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

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
$$f(x) = (\epsilon+|x|^2)^\frac{1}{2}$$

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
$$f(x) = \frac{x_1}{\sqrt{x_1^2 + x_2^2 + ... + x_k^2}}$$

for $1 < k \le d$. 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).

 

[rochfor1]

$$\sum_{ k = 0 }^\infty k \chi_{ [ 0, k^{ -k } ] }$$. It's in $$L_p [ 0, 1 ]$$ for all $$1 \leq p < \infty$$ but not in $$L_\infty [ 0, 1 ]$$.