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This question already has an answer here:

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?

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marked as duplicate by tpg2114, ACuriousMind, John Rennie, Brandon Enright, BMS Aug 20 '14 at 16:11

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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.

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    $\begingroup$ Temperature is only defined for infinite temperature baths, so what you are "measuring" there is not temperature. $\endgroup$ – CuriousOne Aug 20 '14 at 13:52

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