# Discontinuous bijection from $[0,1]$ to $[0,1]$

Is it possible to construct a bijection from $[0,1]$ to $[0,1]$ with odd number of points of discontinuity ?

Yes. Think about graphs; can you draw a bijection with only one discontinuity? As an example, $$f(x)=\begin{cases}x &:\ x\in[0,1/2)\\ 3/2-x &:x\in[1/2,1]\end{cases}.$$
Yes. For example, $$f(x)=\begin{cases}\frac{1}{2} & x=0 \\ 1 & x = \frac{1}{2}\\0 & x=1 \\ x &\text{otherwise}\end{cases}$$
Just draw the graph of $y=x$. Choose any odd no. of points in $[0,1]$ and exchange their values. You are done.
Another solution will be $T$ at an angle of 45 degrees with upper half of its stand deleted.