TITLE: Open maps which are not continuous
QUESTION [51 upvotes]: What is an example of an open map $(0,1) \to \mathbb{R}$ which is not continuous? Is it even possible for one to exist? What about in higher dimensions? The simplest example I've been able to think of is the map $e^{1/z}$ from $\mathbb{C}$ to $\mathbb{C}$ (filled in to be $0$ at $0$). There must be a simpler example, using the usual Euclidean topology, right?

REPLY [14 votes]: There is in fact a rather easy example of a function $\mathbb R \to \mathbb R$ such that the image of every open set is $\mathbb R$: Let $(x_i)_{i\in\mathbb Z_+}$ be the binary expansion of $x$, so that each $x_i \in \{0,1\}$. Let then
$$f(x) = \sum_{k=1}^\infty\frac{(-1)^{x_k}}k\quad \textrm{if the series converges}$$
$$f(x) = 0\quad \textrm{otherwise.}$$
Since the harmonic series (or a tail of it) can be made to converge to any real number by changing signs in the appropriate way, this function has $f((a,b)) = \mathbb R$ for any real $a < b$. Hence this function is open, though clearly not continuous at any point.
The harmonic series can be substituted with any other absolutely unbounded series where the summand goes to zero.