Are temperature increases discrete?

Question

Are temperature increases ever discrete in nature, or is it a continuous variable?

If a discrete case exists, is there any material that exhibits particularly strange behavior?

Answer 1

If you stick with the Statistical Mechanics formalism, then temperature is discrete.

Entropy S is $ S = k_B\ln(\Gamma) $, where $\Gamma$ is the number of available quantum states,

and temperature is

$T = \frac{dE}{dS} $ , where E is the total kinetic energy of the system.

Since every quantity there is discrete, so is temperature.