The function that maps rationals to 1 and irrationals to 0
∅:∅→∅
I found myself pretty fascinated by the gamma function a few months ago... there's something about the extension of factorials to the negative domain that is deeply pleasing to me
The determinant (det: M_n(R) -> R) is king.
I'm going to cheat and say the Dirac delta. It was my introduction to distributions.
Riemann zeta function FTW!
Weierstrass function
Continuous everywhere, but not differentiable anywhere
by
f(x) = x

Identity function is the best function. Change my mind.
Let G be a group, with g in G.

The function G-> G that maps x to gx is my favorite.

The fact that it is bijective (therefore a permutation of G) is the reason why every group is (subgroup of) a permutation group (aka Cayley's Theorem).

That kind of represents Group Theory to me.
Recall that the Cantor set K is the space of infinite bit sequences, topologized as the product of countably many copies of 2. Let L be the space of arbitrary functions K -> 2, and let N* be the one-point compactification of N, i.e., the closure of N as a subset of the real projective line. Define a map w : L -> N* as follows:

* If f : K -> 2 is continuous, then w(f) is the smallest natural number n such that the value of f(x) is completely determined by the first n bits of x. (It is easy to see that f is continuous if and only if such an n exists.)

* If f : K -> 2 isn't continuous, then w(f) is infinity.

Today, my favorite function is w.

**EDIT:** Fixed typo.

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