TITLE: How many non-differentiable functions exist?
QUESTION [18 upvotes]: The size of the set of functions that map $\mathbb{R}\to \mathbb{R}$ equals $(\#\mathbb{R})^{\#\mathbb{R}}$. How many non-differentiable functions are there in this set?

REPLY [48 votes]: Here is a simple way to get the answer:
Suppose a function $f:\mathbb R \to \mathbb R$ is equal to the function $g$ on $\mathbb Q$, where $g:\mathbb Q \to \mathbb Q$ is defined by $g(p/q) = q$ (and we choose the representation $p/q$ so that $q$ is the smallest possible positive integer). Then $f$ is nowhere differentiable, because it is unbounded on every interval.
And the number of such $f$ is $|\mathbb R^{\mathbb R \setminus \mathbb Q}| = |\mathbb R^{\mathbb R}|$, because $|\mathbb R \setminus \mathbb Q| = |\mathbb R|$.
Hence there as many nowhere-differentiable functions $\mathbb R \to \mathbb R$ as there are functions $\mathbb R \to \mathbb R$.
(This doesn't tell you how many differentiable functions there are. The number of somewhere-differentiable functions is the same as the set of all functions; but the number of everywhere-differentiable functions is $|\mathbb R^\mathbb Q| = |\mathbb R|$, because such a function is determined by its values on the rationals.)