Is conduction band discrete or continuous? My question is very simple. I just want to know that is conduction band discrete or continuous?
 A: 
A useful way to visualize the difference between conductors, insulators and semiconductors is to plot the available energies for electrons in the materials. Instead of having discrete energies as in the case of free atoms, the available energy states form bands.


Read on the link.
You ask:

My question is very simple. I just want to know that is conduction band discrete or continuous

The band theory  is a quantum mechanical  model,   such that the difference between energy levels in the band  is very small, mathematically discrete but experimentally continuous. That is why it is called a "band".
A: As other have noted, it's discrete but with fine enough spacing to treat as continuous.
However I disagree that quantum mechanics is the reason. You see the exact same thing in a classical 1-D chain of masses connected by springs.
The allowed wave vectors $\vec{k}$ in the bands are reciprocal lattice vectors, and the number of reciprocal lattice vectors is equal to the number of real-space (aka direct) lattice vectors. That in turn is equal to the number of atoms. So, if you have an infinite number of atoms in your lattice, then there are an infinite number of reciprocal lattice vectors, and your band is continuous. (I guess it's countably infinite which is a little different that continuous, but let's not go there.)
Real materials have a finite number of atoms, so the bands are, strictly speaking, not continuous. However, most crystals that people deal with are large enough that the number of atoms is huge --- effectively infinite.
A: As this model is itself associated with quantum model, so it seems obvious that energy levels occur in steps or we can say that they are discrete but experimental analysis says that they are continuous and it's intuitive that they are too close that's why we call them a band i.e  a band of too close energy levels.
A: To summarize the answers given so far:
In any finite system, the energy levels are discrete, and the number of energy levels is proportional to the number of atoms in your system.
So if you have a particle only a few atoms across, your energy levels are noticeably distinct, even within the same band. But for a macroscopic crystal, the number of levels is in the region of Avogadro's number (about $10^{23}$), so your band will be effectively continuous. Even theoretically, one describes such macroscopic systems by considering the limit of an infinite crystal, so the bands become continuous.
A related phenomenon mentioned in the comments is energy broadening at finite temperatures. This does not change the quantum mechanical energy levels, but prevents you from seeing them as discrete lines in an absorption/emission spectra.
