How does temperature work when a statistical ensemble has discrete energy levels?

Say I’m working with a system which is a collection of N non-interacting spins. For simplicity, I’ll pretend that the energy of each upwards spin is 1, and that downwards spins have no energy. The energy of a microstate is then simply the number of upwards spins it has. Now, since I’m working in the microcanonical ensemble, I’ll declare that my only knowledge of the system is that it has a fixed energy E, meaning the exact microstate of the system is equally likely to be any of the ones which have energy E. I then have the entropy S of my ensemble, which is defined by E, so I have the entropy S(E) of the macrostate with energy E.

As far as I understand, the (inverse) temperature T(E) of this macrostate with energy E is the partial derivative of entropy with respect to the energy, as evaluated at energy E. (Basically, the sensitivity of the entropy of the macrostate to tiny changes in the energy about the value E).

But how does this derivative make sense when I only have a finite, discrete set of values that the energy of my system could take, each of them separated by units of 1. And thus the resulting change in entropy doesn’t make any sense when I make a tiny change in the energy, because I can’t just make a tiny change in energy. I have to go in steps of 1, or else my system isn’t defined.

So… Does this mean that temperature is only defined when the energy of a system can take values in the continuum?