A Riemann integrable, non-regulated function

Hint: Regulated function has left and right limit at every point.

The function $f:[0,2] \to \Bbb{R}$ $$f(x) = \left\{ \begin{matrix} 1 & \mbox{ if } x=1+\frac{1}{n} \mbox{ for some $n \in \Bbb{N}$} \\ 0 & \mbox{otherwise} \end{matrix}\right.$$ is Riemann integrable, but it is not regulated.