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Photon is the quantum of light, i.e., energy comes in discrete level. But EM energy is a function of frequency ($E=hf$). And if frequency has a continuous spectrum, then energy level is continuous, instead of discrete.

Doppler Effect tends to prove the argument of continuous frequencies. Space is continuous, even though any length less than Planck's length has no meaning. As stars and galaxies move in a continuous space, then Doppler continuously shifts (blue and red) the frequency of their radiated EM wave infinitesimally.

The logical question would therefore be: is frequency continuous? Meaning, do we have frequencies like 1.012345 Hz, 100.098765 Hz, etc?

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Yes, frequency is continuous. Energy and momentum of photons is continuous.

Atoms have electrons in discrete energy levels. Photons are emitted when an electron drops from an excited state to a lower state. This process emits photons at discrete energy levels, where each type of atoms has its own set of energy levels.

Typically, discrete energy levels arise when a particle is confined to an energy well.

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