For a research project I work with continuous distributions, like the normal distribution. In my use case however the random variable Z generally follows a normal distribution, though it can only take discrete values (..., -3, -2, -1, 0, 1, 2, 3, ...)$\in \mathbb{Z}$.

For the "discretization" I use a simple heuristic. An easy example would be a standard normal distribution:

$$ Z \sim \mathcal{N}(0,1) \\ f(Z) = \sum_{i\in \mathbb{Z}} \mathbb{1}_{(i-0.5, i+0.5]}(Z) \\ X = f(Z) $$

My question now is: Is there a special name for the resulting distribution X? Is there any research on the resulting distribution?

Best, Simon