[jdhwosnhw]

I have yet to see differential entropy used successfully (beyond its explicitly constructed-for purpose for calculating channel capacity). Similar to your thought experiment is the issue that the differential entropy value depends on your choice of unit system. Fundamentally the issue is that you cant stick a quantity with units into a transcendental function and get meaningful results out

[dynm]

Yeah, it's quite disturbing that the differential entropy (unlike the discrete entropy) depends on the units. Even worse, the differential entropy can be negative!

Interestingly, the differential KL-divergence (differential cross-entropy - differential entropy) doesn't seem to have any of these problems.