Question 78
Let $f: \mathbb{R} \to \mathbb{R}$ be a function such that, for every $x, y \in \mathbb{R}$:
$$
|f(x + y) + \sin x + \sin y| \le 2.
$$
a. Prove that $|f(x)| \le 1 + \cos x$, for every $x \in \mathbb{R}$.
b. Give an example of such a function, which vanishes nowhere in the interval $(-\pi, \pi)$.
$$
|f(x + y) + \sin x + \sin y| \le 2.
$$
a. Prove that $|f(x)| \le 1 + \cos x$, for every $x \in \mathbb{R}$.
b. Give an example of such a function, which vanishes nowhere in the interval $(-\pi, \pi)$.