This question is about the so called 'branch cut' for defining multivalued function continuously on complex plane.For example to define inverse of the function $f(z)=z^2$ we consider the domain of the inverse function as the slit plane $C\setminus(\infty,0]$. The case is also similar when defining Log function.
So my question, is there any sort of 'rule of thumb' which guarantees that removing this sort slit assures you that the domain is right for defining inverse function? Also why we are removing straight lines always? Would we get same results if we have removed any arbitary type of curves from C?