Miklos Schweitzer 2013 Problem 8
Problem 8. Let be a continuous and strictly increasing function for which
for all ( denotes the inverse of ). Prove that there exist real constants and such that for all .
Solution: Pick . Then we have
Using the hypothesis we find that is strictly monotone on a a finite number of subintervals (depending on the sign of and ) so as a consequence can take only a finite number of values. Since is continuous we conclude that for every the , a real constant. As a consequence we have
for every .
Let and consider a positive integer . Denote . Then
so and . Pick now a positive integer and notice that
so we have . This proves by continuity that for we have .
Using the identity we can extend the result for negative and we are done.
Advertisements
Categories: Algebra, IMO, Olympiad, Problem Solving
functional equation
Comments (0)
Trackbacks (1)
Leave a comment
Trackback

November 9, 2013 at 2:43 pmMiklos Schweitzer 2013  Beni Bogoşel's blog