The spectrum from a black body comprises discrete emissions over the full light spectrum. Atoms have characteristic emissions from the electronic shells. A solid has emissions over the full spectrum. The temperature of the solid determines the peak of the spectrum. The emissions from a heated solid come from individual atoms and interactions between and among atoms of the solid. There are many vibrational modes for atomic interactions that produce the spectrum of emissions. The black body spectrum is theoretical and is well approximated by some materials and shapes. Most solids are not good black-body emitters.

Dear Kirtikesh Kumar,

in my opinion you should distinguish between two things.

(1) There is the spectral theory, according to which a physical observable, such as energy, can have both, a discrete as well as a continuous spectrum (https://en.wikipedia.org/wiki/Decomposition_of_spectrum_(functional_analysis)). In this case the mathematical spectrum shows the expectation values of the energy operator (or of any other operator).

Stated briefly, the continuous part of the energy spectrum is associated with black body radiation, by which is meant the thermal radiation emitted by a cavity at a very high fixed temperature, whereas the discrete spectrum is associated with atomic electron transitions from a bound state energy level to another. Ionization energies belong to higher energy ranges (https://en.wikipedia.org/wiki/Ionization_energy).

(2) There are spectroscopic emission data. In that case a gas (or a material in another state of aggregation) is excited so as to radiate, and the term spectrum refers to the acquired response in some wavelength range. In the visible range one normally can detect from broad to quite narrow emission bands. For example, the latter can be obtained from partially evacuated electrical discharge tubes (https://en.wikipedia.org/wiki/Gas-filled_tube), and there is a non negligible contribution to the measured bandwidth from the detector. Bandwidths also depend on the circumstances of acquisition. Under some circumstances patterns can be detected in the bands (for example http://www.jstor.org/stable/pdf/83413.pdf?acceptTC=true --On Thiele's 'Phase' in Band Spectra.)

I think that if you read the papers trying to derive black body radiation, and the one finally explaining its spectrum properly by including quantum effects, it will answer your question.  The things you are asking about were considered 100 years ago and are included in the theory.  Atoms emit discrete spectra but these are spread, for instance, by Doppler from thermal movement, by frequency changes due to the environment the atom is in, i.e. which molecules they are are in, and by other things.  I expect that there are so many discrete lines and that they are all spread or widened so that the spectrum is continuous.  If you read the original papers they probably deal with all this.  It would be surprising if a theory that fits the data so well has missed out something so significant and nobody has noticed for 100 years.

Dear Kirtikesh,

That is a question that has been bothering me too. I think that the answer is that blackbody radiation is a form of Bremsstralung (braking radiation.) Bremsstralung also produces a continuous spectrum because it is produced by a continuous process. As an electron approaches the positive nucleus of an atom it loses kinetic energy in a continuous fashion. The electrons in a metal will vibrate with increasing intensity as the temperature increases and so would produce more intense continuum radiation as they interact with the nucleus.

Malcolm writes that " It would be surprising if a theory that fits the data so well has missed out something so significant and nobody has noticed for 100 years." However, the idea that energy is quantised was first proposed by Einstein, although Planck had shown that it is action that is quantised. No one dares to argue against Einstein!

Max Plank found that assuming quantization of emission of radiation explained the shape of the black-body radiation curve in 1900 (other theories gave high radiation at high frequencies), and Einstein explained the photoelectric effect in terms of quantization of received radiation in 1905.  Lots of people have dared to argue with Einstein (and Plank, and Schrodinger), and it's a good thing they do, but no-one has managed to come up with an argument that explains experimental results better.

The experimental results that are so good are for line radiation e.g. the Lyman and Balmer serieses. No one has come up with an answer to Kirtikish's question because those results were so good. Line and blackbody radiation are two different things and need different explanations. Are you saying Bremsstrahlung does not exist?

Max Planck found a formula which matched the experimental values for black-body radiation, but he then had to find why the formula worked. His conclusion was that there was there was a quantum of action. The quantum of energy is a result of action being quantised.

 Experimental results are equally good for black body radiation as line radiation. The problem arises when attempting to model the experimental results. Planck found a model that fit the experimental results. The conceptual model is that black body radiation appears continuous, but the mathematical model fits only when it is assumed discontinuous.

Modeling often requires empirical constants or variables that eventually receive names. Sometimes naming does not constitute explaining. Electricity/magnetism and thermodynamics have many such examples. Planck's constant resulted in an explanation that has withstood over a century of experimentation.

Black body radiation is modeled by a continuous curve, but the model is not the process. This is the answer to the original question. Planck's equation is the model not the process.

Bremsstrahlung is entirely different as to source and generation. A bremsstrahlung energy spectrum is considered continuous, but comprises the energy from discrete events.

I am unaware of any evidence that in reality Plank's curve is discontinuous.

What is the quantum of frequency? Bremsstralung is produced  by the deceleration  of electrons. How  is that quantised?

 A photon is a single quantum of light. The energy of a photon can be expressed as frequency or wavelength. Theory assumes that there is a limit to length, the quantum length or 1.616229(38)×10−35 m. The blackbody and bremsstralung spectra expressed as wavelength are discontinuous, but not so that you would notice.