Why we need branch cut?[Continuity and analytic on Branch cuts] Why we need branch cut in Complex analysis? It just my guess though, complex function itself is multi-valued function, so To define the one-valued function we take the branch cut, right? 
Also what about the continuity and analytic on the branch cut for the function $f(z)$ for $z \in \mathbb{C}$? Is $f(z)$ always both discontinuous and non-analytic on branch cut and the branch point? (The below e.g. is reason that why I did think that.)
e.g.) $f(z) = z^{1\over2}$ for $D = \{z \vert -{3\pi\over2} \leq arg(z) \lt {\pi\over 2} \}$ (It is not analytic on the $\{z \vert arg(z) = -{3\pi\over2} \}$ )
Plus Does entire function not have a branch cut?

You are right: "To define the one-valued function". Branch cut has to start at branch point and then goes arbitrary path to infinity. In this way Riemann surface is cut into Riemann sheets. On a given sheet function is on the cut discontinuous and non-analytic. Of course, function remains analytic if you go from one sheet to the next one. It has to be non-analytic at branch point because the Taylor expansion (as criteria for analyticity) cannot take "multiple-valued" character of the function into account, Taylor expansion is always single-valued. The same argument applies for cut. Imagine the function on one single Rimmann sheet would be analytic at cut: how it would be continued to a different sheet? There would be only one sheet and no cut.
By definition entire function have no cut neither branch point.
