Tessellate S.T.E.M.S (2019) - Mathematics - Category C - Set 3 - Objective Problem 3

Let $f:\mathbb{R} \to \mathbb{R}$ be a continuous function and $a$ be a real number such that $\lim_{x\to\infty}f(x) = \lim_{x\to-\infty}f(x) = a$

Let $S = \{d \mid \exists x , y$ such that $x - y = d \&\ f(x) = f(y) , d\in \mathbb{R}\}$ . Which of the following is a set that can be equal to $S$