Is frequency quantized in the black body spectrum? I'm aware that there're some questions posted here with respect to this subject on this site, but I still want to make sure, is frequency quantized? Do very fine discontinuities exist in a continuous spectrum like the black body spectrum?
The quantization of photon energies
 A: "Yes", but the quantisation depends on the size of the box. In practice the 'box' is large and of variable shape, so all sizes are available, so all frequencies are available. 
Ultimately, it is somewhat of a philosophical question who's answer depends on which axioms and base concepts you (they) are using at the various stages of reasoning. Consider, does time pass for a non-interacting particle? Can a non-interacting particle be kept in a box? etc. 
Try for a quirky view on the problem. Link Between the P≠NP Problem and the Quantum Nature of Universe

A: Frequency is not quantized, and has a continuous spectrum. As such, a photon can have any energy, as $E=\hbar\omega$. However, quantum mechanically, if a particle is restricted by a potential, i.e.
$$\hat{H}=-\frac{\hbar^2}{2m}\nabla^2 + \hat{V}$$
for $V\neq 0$, the energy spectrum is discrete. For example, in the case of the harmonic oscillator,
$$E_n=\hbar \omega \left( n+\frac{1}{2}\right)\quad n=0,1,2,\dots$$
A: well, the frequency of a photon is not quantized, we observe all 'kinds of colors (frequencies)'...what is quantized is the quantity of photons of the exact same frequency we may produced, or otherwise exist...for example, lets say we have a photon of 633 nm, then the quantity of photons that may be 'produced' is 1, 2, 3, 4, 5,because the photon is quantized, or the energy of a photon of an exact frequency can be increased only by steps of hν,  here ν is the not quantized frequency....quantum physics is pretty simple.
