Zero variance Random variables with density

Question

I found here the question:

Can a random variable have a density function whose variance is $0$ ?

I understood as a random variable which has a density.

What is your opinion on what I understood and if that is the case on the answer that was given here

Answer 1

Well, to the extent that you let the Dirac delta function act as a density, then yes. An RV with zero variance is almost surely constant, so its just a degenerate discrete random variable.