I'm a little temporarily confused about the concept of differential entropy. It says on wikipedia that the differential entropy of a Gaussian is $\log(\sigma\sqrt{2\pi e})$. However I was thinking as $\sigma \rightarrow 0^+$, the only intuitive value for an entropy to me seems to be 0. We are then 100% sure that the outcome will be equal to $\mu$, and nothing is required to store the knowledge of what the outcome will be. Instead the expression above gives $-\infty$.
So I must be misunderstanding something, right?
Just to clarify, the reason why I'm asking is that I'm trying to figure out if my approach at this question about Empirical Entropy makes any sense.
Edit: "Own work"
Now I have thought a bit about this. If we take the easiest distribution, the uniform distribution, which (according to wikipedia) has differential entropy $\log(b-a)$, say $b-a = 2^k$.
If k = -1, this would be -1 bit and the interval would be length 0.5.
If k = 0, this would be 0 bit and the interval would be length 1.0.
If k = 1, this would be 1 bit and the interval would be length 2.0.
So if the interval is 0.5, we would "save" one bit, as compared to if we had to store the precision of an interval of length 1. So differential entropy is in some sense the information needed "in excess" to whatever resolution we want to store with. Does this make any sense?