Abel's theorem in problems and solutions - School of Mathematics

178 Problems of Chapter 2
FIGURE 66
285. Let C be a curve not traversing the cut and joining the points
and Suppose that by choosing two distinct values at the point and
defining the function by continuity along the curve C we obtain the
same value at the point Consider thus the curve i.e., the curve C
oriented in the opposite way. We obtain that the value at the initial point
of is the same in both cases, but the values at the final point
defined by continuity, are different. This is not possible by virtue of the
uniqueness of the continuous image, because the curve C does not pass
through the point It follows that our claim that is not
true.
FIGURE 67 FIGURE 68
286. Let be an arbitrary point outside the cut, and let be a
continuous curve starting from and ending at without crossing the
cut. Let us draw another curve not crossing the cut, going from the
point to the point (Figure 67). By hypothesis we have chosen the
value This means that if we choose and we define
by continuity along the curve we shall obtain exactly But
thus the value of defined by continuity along the curve according