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

168 Problems of Chapter 2
corresponds to a dilation by simultaneous with a rotation by the angle
(Figure 59).
247. When varies continuously from 0 to 1, varies continuously
from 1 to 0. The function thus describes the same
geometrical curve as C, but oriented in the opposite way.
248. When varies continuously from 0 to varies continuously
from 0 to 1. When varies continuously from 1/2 to 1, varies
continuously from 0 to 1. Hence the function given by the problem
describes the curve which is obtained by drawing first the curve and
then the curve The condition guarantees the continuity
of the obtained curve.
249. Answer.
250. Answer. a) b) c) d)
251. For every we have where the values of
may be distinct for all We thus write It follows
that Since the functions and are
continuous for the function is also continuous for
(cf., 239, 240). But since the function takes only integer values it is
continuous only if it is a constant, i.e., if where is some given
integer which does not depend on Therefore
and
252. Let and be two functions describing the continuous
variation of and Thus (cf., 251)
where is a given integer. But hence
Consequently and
253. Let and be two functions describing the continuous
variation of Thus (cf., 251) where is a given
integer. In particular, Consequently
and
254. a) One can take Answer. (see Figure
53); b) Answer. (see Figure 52); c) Answer.
d) Answer. (see Figure 60)
255. Answer. a) b)
256. Answer. a) 1; b) –2; c) 2; d) 0.