MathDB

f(x + a) is even, and f(x + b) is odd => f(x) is periodic

Source: 2015 Brazil IMO TST 3.1

July 23, 2021

functionoddEvenperiodicalgebra

Problem Statement

Let's call a function

f : R \to R

cool if there are real numbers

a

and

b

such that

f(x + a)

is an even function and

f(x + b)

is an odd function. (a) Prove that every cool function is periodic. (b) Give an example of a periodic function that is not cool.