Let's say that I have a single point at $(0.5,0.5)$.
The single data point is uniformly distributed in the unit square so I would sort of expect it to have zero discrepancy. (related: How to calculate discrepancy of a sequence)
However, if we calculate it, we get a non zero result, because for instance the half open region $[(0.0, 0.0), (0.5, 0.5))$ contains no points and has an area of 0.25.
Is it expected that a uniformly distributed data set wouldn't have a discrepancy of zero?
My best guess is that yes this is expected because discrepancy compares area vs density which are both continuous values, while specific data points are discrete.
Is this correct / the correct thinking?