How is a continuum spectrum possible?

Question

Considering the fact that energy is quantized, shouldn't the spectrum of light also be quantized?

Answer 1

Black body continuum radiation is the result of excitations of the electrons of atoms and molecules by thermal collisions. Because of thermal motion of the atoms or molecules the photons emitted due to transition between levels separated by an energy hv, will have some spread, (line width). If the photon is absorbed and re emitted many times in the volume along the path to the emitting surface the line broadens, and if it happens enough times the lines will merge to form a continuum.
The standard way to construct a "block body" is a cavity with an absorbing surface, kept at constant temperature, with a small hole for the radiation to emerge. This assures that the emerging photons will have had many absorptions and emissions before exiting.
Since the energy transitions in molecules are between rotational and vibrational levels, there are many closely spaced lines in the emissions spectrum to start with.

Answer 2

Only the bound states are quantized. If an electron passes a charged nucleus with too large an energy to be automatically bound, but nevertheless is captured, it is going into a bound state from an unbound state, and any energy of radiation can result. Thermal spectra are an example of this.

Answer 3

While light is quantizingly dispersed (using photons), it travels in waves. It is these waves which exhibit a continuum of frequencies.