奇怪的函数

It's important to realize that a function represents any map between two sets. Some examples of functions (let's say from reals to reals) include $$f(x) = \begin{cases} q^{-1} & \text{if $x=\frac pq$ is rational, and $\gcd(p,q) = 1$} \\ 0 & \text{otherwise}. \end{cases}$$ This particular function has some interesting properties (for example, it is continuous at $x$ if and only if $x$ is irrational). Another example of a pathological function with interesting properties is the Weierstrass function. Note that functions don't even need to have closed forms!

In other words, unless otherwise specified you should assume nothing about a function other than it gives one output for every input. In particular, you should not assume it is continuous, differentiable, that it has a "closed form", or even that it can be plotted. (See the first paragraph of this section to see how bad the situation can get.)

Here's a fun exercise with a stunning answer. Find the "strangest" function $f : \mathbb R \to \mathbb R$ you can such that $$f(x^2+y^2+2f(xy)) = f(x+y)^2.$$ Of course $f(x) = x$, $f(x) = 0$, and $f(x) = 1$ are three such functions. However, much stranger functions exist!

Here's the entire list of functions which work. Aside from the three above, we also have $$f(x) = \begin{cases} -1 & \text{if $x \in S$} \\ 1 & \text{otherwise} \end{cases}$$ where $S$ is any subset of the interval $\left(-\infty, -2/3\right)$.

Given what we just said about how general functions are, it might be surprising that we can prove these are the only such functions! In fact, this happens to be IMO Shortlist 2004, Problem A6, a notoriously hard olympiad problem.

If you're interested, these types of problems (finding all functions that satisfy some condition) are called functional equations.