TITLE: Is there a bijective function $f: \mathbb{R}\to\mathbb{R}$ that is discontinuous?
QUESTION [5 upvotes]: Is there a bijective function that is discontinuous?

REPLY [12 votes]: Discontinuous everywhere: $f(x)=x$ if $x$ rational, $x+1$ if $x$ irrational.

REPLY [7 votes]: $$f(x) =\left\lbrace \begin{array}{ll}
x & \text{ if } x\not\in\lbrace 0,1\rbrace \\
1 & \text{ if } x=0 \\
0 & \text{ if } x=1
\end{array}  \right.$$
for example.