I have heard people say that temperature is basically continuous (the temperature can’t jump from 10 degrees to 30 it gradually increases) is this true at a quantum level and if not what is the smallest discrete amount temperature can change
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Continuity is a mathematical concept, a possible property that some mathematical object might have. Mathematical objects only exist in human minds.
That said, temperature is a very useful mathematical abstraction in physics, and treating it as continuous leads to results that agree with experiment. But there is no "truth" here in a mathematical sense.
John Doty
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Temperature is a bulk property, it's not a property individual atoms have, by definition. It relates to average kinetic energy within a material relative to it's edges or container.
That said an individual atom in a very small container would still emit thermal photons, & it should be possible to assign it a temperature if it can be regarded as a black body, though it would only be valid at the moment of photon emission.
Temperature relates to internal degrees of freedom, for instance hydrogen can exchange rotational energy around the axis of the two atoms, but helium cannot.
CriglCragl
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