How was the emission of lights of atoms explained in Classical Electromagnetism? In QM, it is explained that atoms emit light by 'jumping' down orbitals, losing energy, and therefore releasing light of a certain frequency. In Electromagnetism, before QM was invented, how was this emission of light explained? It is true that their explanation broke down for high frequencies, and that this difference in prediction and experiment (Blackbody Radiation/Ultraviolet Catastrophe) led to QM's invention, but this means for lower frequencies their explanation, to some extent worked. What was this explanation, based on continuous, non-quantized, light?
In essence, how is the emission of light from atoms explained in Electromagnetism?
 A: The ultraviolet catastrophe was a consequence of statistical mechanics, agnostic as to the radiation mechanism.
Hertz had demonstrated that oscillating electric currents radiate electromagnetic waves. There was abundant evidence (electrochemistry, cathode rays, ...) that there was electricity inside atoms. In Thomson's plum pudding model, oscillations of electrons within the pudding potential could radiate, but they couldn't account for atomic spectra. But Bohr's model captured the spectrum of hydrogen, and quantum mechanics took off.
A: Your description of electrons "jumping down orbitals and emitting light" is incomplete. What does it mean to "jump down an orbital"?
In both the classical and quantum cases atoms emit light due to the coupling of the electrons in atoms to the electromagnetic field. In both the classical and quantum cases when a charge is accelerated (like an electron in an atom) the charge loses kinetic energy and the electromagnetic field is excited (i.e. the amplitude of the electromagnetic field increases in the vicinity of the accelerating charge).
The classical description failed largely because it predicted atoms would emit radiation until the electron collides with the nucleus at which time the atom would be essentially inert. This was inconsistent with experiments showing atoms had finite and unchanging spatial extent.
Your description of the ultraviolet catastrophe is a bit unrelated to the failure of classical mechanics to describe radiation of light by atoms. That's a different story as explained in John Doty's answer.
A: Emission and absorption of light by atoms can be treated by modelling the atom as an oscillating electric dipole and analysing its interaction with an electromagnetic wave.
The model actually explains reasonably well the radiative lifetimes and absorption cross-sections of many transitions, but has numerous points of failure.
It cannot explain why atoms in the ground state don't just radiate until the electrons spiral into the nucleus. It cannot explain why some transitions are much less likely than others (forbidden transitions) and it cannot explain really explain why spontaneous emission occurs in the absence of a stimulating electromagnetic field.
