Discontinuous bijection example Is there an example of a function that is discontinuous but bijective whose inverse is also a bijection and discontinuous? 
 A: $$f(x) = \begin{cases} x & \text{if }x\in \mathbb Q \\ x+1 & \text{if }x\in \mathbb R\setminus\mathbb Q \end{cases} $$
Or even
$$ g(x) = \begin{cases} -x & \text{if }x\in\mathbb Q \\
 2\sqrt2-x & \text{if }x-\sqrt2 \in \mathbb Q \\
x & \text{otherwise} \end{cases} $$
which is its own inverse and nowhere continuous.
A: A very simple pictorial example:

A: The inverse of a bijection is always a bijection.
The inverse of a function that is not everywhere continuous is in some cases everywhere continuous, and in particular there exist bijections that are not everywhere continuous but whose inverse is everywhere continuous, so there's more work to do than just finding a bijection that is not everywhere continuous.
How about this:
\begin{align}
f &: [0,1)\to [0,1) \\[5pt]
f(x) & = \begin{cases} x+\frac 1 2 & \text{for } 0\le x\le \frac 1 2, \\[5pt]
x - \frac 1 2 & \text{for } \frac 1 2 \le x < 1. \end{cases}
\end{align}
This is an involution, i.e. it is its own inverse. It has a jump discontinuity at $x=\frac 1 2.$
