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

164 Problems of Chapter 2
If then the inequality holds for all
and thus can be chosen arbitrarily. If consider on the plane
with coordinates and a circle of radius 1 with centre at the origin of
the coordinates and draw the straight lines and (Figure
49). For the angles and shown in the Figure we obtain
Hence
FIGURE 49
Choose Thus from it follows that
i.e., Hence
i.e., and Thus
Consequently the functions and are continuous
for all the real values of the argument
243. Let a point and an arbitrary positive real number be given.
We have to choose a real number such that for every satisfying
the inequality (and, of course, the inequality
holds. This last inequality is equivalent to the inequalities
and