What is your favorite "strange" function?

The empty function $\emptyset:\emptyset\to\emptyset$ is quite strange when you first meet it.

The Busy Beaver function

Let Σ be a finite alphabet, for instance {0, 1}; let M be the set of Turing machines with alphabet Σ, and let H ⊆ M be the set of Turing machines that halt when given the empty string ε as input.

For each M ∈ H, Let s(M) be the number of steps performed by M before halting (when given ε as input).

Finally, let S : ℕ → ℕ be the function defined by

S(n) = max {s(M) : M ∈ H and M has n states}

Notice that S is well-defined, since only finitely many Turing machines with n states exist.

In other words, S(n) is the maximum number of steps performed on ε among all halting Turing machines with n states. S is called the Busy Beaver function.

It turns out that S is uncomputable because it grows faster than any computable function, that is, for all recursive functions f : ℕ → ℕ we have S(n) > f(n) for large enough n, and in particular f is o(S).

A Brownian motion sample path.

These are about the most bizarrely behaved continuous functions on $\mathbb{R}^+$ that you can think of. They are nowhere differentiable, have unbounded variation, attain local maxima and minima in every interval... Many, many papers and books have been written about their strange properties.

Edit: As commented, I should clarify the term "sample path". Brownian motion is a stochastic process $B_t$. We say a sample path of Brownian motion has some property if the function $t \mapsto B_t$ has that property almost surely. So, run a Brownian motion, and with probability 1 you will get a function with all these weird properties.