Measuring the position and momentum of a particle is not simultaneously possible according to Heisenberg's uncertainty principle. Heisenberg's uncertainty principle gives us the uncertainty in measuring the position and momentum of a particle simultaneously. We can think of this as the highest possible resolution in measuring distance as $\Delta{x}$, and highest possible resolution in measuring momentum as $\Delta{p}$. So momentum and position can be thought of as varying in units $\Delta{p}$ and $\Delta{x}$ respectively, so they can't take on continuous values.
This is my interpretation of Heisenberg's uncertainty principle. My interpretation is not only we cannot measure position and momentum simultaneously, but that momentum and position take on discrete values. For position, it would be $x+\Delta{x}$, $x-\Delta{x}$, $x+2\Delta{x}$, ..., $x + n\Delta{x}$, where $n$ is an integer. This implies discretization. Is my interpretation right?