 In calculus, we define continuity as the limit of a function is equal to the function value provided that both sides exist. But in complex analysis, we must define continuity this way, and you should pay careful attention to the significant difference between the two. Now, if we view our function as a sum of real and complex functions of two variables, then proving the continuity of a complex function is like proving continuity for multivariable functions. It's very hard to do. But in general, if f is an algebraic or transcendental function of z, f of z is continuous for all z where it is defined. So our primary concern is going to be about functions like real part of z or argument of z, and for those we have to verify or refute that the limit is the function value. For example, let's try to prove that real part of z is continuous for all z. So we want to check to see if the limit as z approaches some z naught of the real of z is the same as the real part of z naught. So let z equal x plus i y, z naught, x naught plus i y naught, then f of z is just x, and f of z naught is x naught. Now, as z approaches z naught, x approaches x naught, and y approaches y naught. Why not? Because it has to. And that means that f of z approaches x naught. And since that is f of z naught, then real part of z is continuous for all z. The imaginary part of z and the modulus of z are also continuous for all z, and we can prove those using essentially the same argument. Unfortunately, the argument itself poses a problem for positive real values of z. So remember, the value of a limit is independent of the path to the point, but if we have a real value of z, if we approach from above, the argument tends to zero, while if we approach from below, the argument tends to pi. And because there's a disagreement, we would say that the argument of z is discontinuous for real values of z. Now, we don't really like that, so let's see what we can do to fix it. Suppose z is cis theta, then z is cis theta plus 2 pi as well. Since we want functions to have a single output, we have to choose a principal value for the argument of z. In this case, we do that by restricting our output of argument z to the interval between zero and 2 pi. This means that points that appear close, like these two, might not be close. To represent the situation where points that appear close are not in fact close, we use a branch cut. We can think of this as a do not cross line in the domain. For the argument of z, we use the positive real axis. For points not of the branch cut, our z will be continuous because we can find a small enough neighborhood that avoids the branch cut. We'll talk more about branch cuts later, as they play an important role in integration.