It depends on what kind of entropy you are computing. If it is an entropy of $N$ particles occupying a $M$ number of energy levels, then your multiplicity will be taking discreete values, and your entropy as well. If you consider magnetic particles, which have two- or three-dimensional magnetization vector, then you Hamiltonian, your partition function, and, hence, the entropy will all be continuous.

It depends on what kind of entropy you are computing. If the problem is computing an entropy for $N$ particles occupying a $M$ number of energy states, then your multiplicity will be taking discreete values, and your entropy will be discrete as well. 

If it is a problem of computing entropy for $N$ particles distributed in $(\vec{x}, \vec{p})$ phase space, multiplicity will be continuous, and the entropy will be continuous too.

Finally, if you consider magnetic particles, which have two- or three-dimensional magnetization vector, then you Hamiltonian, your partition function, and, hence, the entropy will all be continuous.

It depends on what kind of entropy you are computing. If it is an entropy of $N$ particles occupying a $M$ number of energy levels, then your multiplicity will be taking discreete values. If you consider magnetic particles, which have two- or three-dimensional magnetization vector, then you Hamiltonian, your partition function, and, hence, the entropy will all be continuous.

It depends on what kind of entropy you are computing. If it is an entropy of $N$ particles occupying a $M$ number of energy levels, then your multiplicity will be taking discreete values, and your entropy as well. If you consider magnetic particles, which have two- or three-dimensional magnetization vector, then you Hamiltonian, your partition function, and, hence, the entropy will all be continuous.