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|f(x+y+sinx+siny)|<=2

Source: Romanian NO 2011, grade x, p.1

October 3, 2019
functionalgebra

Problem Statement

Let f:R⟶R f:\mathbb{R}\longrightarrow\mathbb{R} a function having the property that ∣f(x+y)+sin⁡x+sin⁡y∣≤2, \left| f(x+y)+\sin x+\sin y \right|\le 2, for all real numbers x,y. x,y.
a) Prove that ∣f(x)∣≤1+cos⁡x, \left| f(x) \right|\le 1+\cos x, for all real numbers x. x. b) Give an example of what f f may be, if the interval (−π,π) \left( -\pi ,\pi \right) is included in its [url=https://en.wikipedia.org/wiki/Support_(mathematics)]support.
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