I know its number of states per unit energy but what happens in the case of continuous energy?
The density of states at an energy E in the continuous case is the amount of states between the energies E and E+dE. It's a distribution, much like a continuous probability distribution. In a continuous probability distribution the probability of any SPECIFIC event is 0 since the probability is 1/N (where N is the number of possibilities, which is infinite in a continuous distribution). It's only in the context of an integral OVER A RANGE of possibilities that you get a finite probability. Same with the density of states, in the continuum limit it can really only be interpreted within an integral.
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