While doing some exercises on complex functions, I noticed that every time I find a branch cut in the complex plane it is always in between two branch points, such as in the function
\begin{equation} f(z) = \sqrt{z} \quad \text{for }\; z\in\mathbb{C} \end{equation}
where there are two branch points, in $0$ and at $\infty$, and there's a branch starting a the origin and then going to $\infty$.
Another example could be the function
\begin{equation} g(z) = \sqrt{(z+1)(z-1)} \quad \text{for }\; z\in\mathbb{C} \end{equation}
where the branch points are in $\pm 1$ and there's a branch cut connecting the two of them. Is this always true?
If that's the case, can the inverse be said, meaning that given two branch points there's always a branch connecting them? And if this is true, what am I to do in the event of a function having more than two branch points? Does this also mean that there's no function with a single branch point? (Note that I've never encountered such a function, so I suspect that this is true)
Also, note that I know how to figure out if there's a branch in a certain area of the plane, I'm asking whether this things can be affirmed prior to calculations.
Sorry for the number of questions but I thought it would've been appropriate to ask them all in the same thread. There's an example of connecting 4 branch points in a comment in this thread, even though it might not be the same.