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Atomic spectroscopy supports the idea of light, not electrons, as particles (though electrons are also particles) because photons must match the specific energy levels of the atom (or molecule) in order to be absorbed/emitted. Otherwise, the light simply passes through.
Atomic Spectroscopy supports the idea that light functions as a particle. This is demonstrated by the fact that using a higher intensity of a low energy light source does not cause electrons to be ejected, which should have happened if light is acting as a wave. Instead, a high energy light source must be used because individual photons are interacting with individual electrons, thus light is behaving as a particle in that experiment.
Atomic spectroscopy falls under the experiments that support quantum mechanics. I think you meant "photons have either wavelike or particle-like properties." When a source of light has the energy matches/exceeds the energy difference between energy levels, it will excite the electron and move it to the next level; otherwise, the photon will pass through. This happens in discrete amounts meaning that there is no "in between" energy level (n=1.1, 1.2, 1.3,...), only n=1, 2, 3, and so on. Therefore, it is not "continuous" (wavelike properties).
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