Confusion About Energy Bands I read about energy bands in a physics textbook, and I couldn’t understand one thing about them. The book defines an energy band as continuous energy levels of electrons that are so plentiful that they are like a continuous band. But, isn’t it impossible for energy levels of electrons to be continuous because of quantum principles, namely, don’t energy levels have to be quantized? Please clarify.
 A: No, they don't.  The allowed energies for a quantum mechanical system comprise the spectrum $\sigma(H)$ of the system's Hamiltonian operator $H$.  This spectrum may consist of discrete points as in the quantum harmonic oscillator where $\sigma(H_{QHO})=\{(n+\frac{1}{2})\hbar\omega\}$, one or more continuous intervals as in the free particle where $\sigma(H_{free})=[0,\infty)$), or a mix of both as in the hydrogen atom where $\sigma(H_{hydrogen})=\{-\frac{13.6\text{ eV}}{n^2}\} \cup [0,\infty)$. As per your textbook, the spectra of periodic Hamiltonians is generally composed of continuous intervals called bands separated by gaps.
A: Quantization does not exclude the possibility of intervals of possible values for the energy. Even without going to the energy bands in a solid, the quantized Hydrogen atom has a discrete part of the spectrum of the Hamiltonian but also a continuous part (unbound scattering states). After completing my answer, I see that there is J.Murray's answer mentioning such a possibility.
However, there is a difference between the continuous part of the spectrum of atoms and the case of the bands of energy in a crystalline solid. In the former, the continuous spectrum is a direct consequence of the analytical properties of the Hamiltonian operator. In the latter, a finite periodic boundary system would have, in many cases, a finite number of isolated eigenvalues of the Hamiltonian, labeled by wavevectors ${\bf k}$ (Bloch's theorem). However, in the limit of an infinite system, the wavevectors become dense, and one obtains a continuum interval of values.
A: Energy level don't need to be quantized (discretized) in quantum mechanics. For example, Hydrogen atom Hamiltonian can hold continuous eigenvalues.
See:https://9to5science.com/continuous-spectrum-of-hydrogen-atom.
A: 
But, isn’t it impossible for energy levels of electrons to be continuous because of quantum principles, namely, don’t energy levels have to be quantized?

Yes, energy levels have to be quantized and can belong to one electron at a time. The term "band"  describes the dense energy levels which can occur when solving the quantum mechanical conditions.
Take the simple hydrogen atom:

One sees that there is a dense "band"  that a single  electron can occupy before the ionization.
In complicated boundary conditions as  with crystal and lattices in general, "bands" means that an individual electron can occupy one level and another electron another level so close in energy to it that it can be considered the same in energy, within measurement errors . In a solid there are order of $10^{23}$ atoms with their electrons, forming a lattice, and within measurement errors the available energy levels for the outer electrons can be seen to be continuous . Each electron will be occupying a single energy level, it is collectively that they occupy a "band".
This has led to the band theory of solids, a quantum mechanical model that allows the study of materials.
Each individual electron occupies one energy level in  the band.
