APMO 1994 Problem 1

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that (i) For all $x, y \in \mathbb{R}$,

$$f(x)+f(y)+1 \geq f(x+y) \geq f(x)+f(y)$$

(ii) For all $x \in[0,1), f(0) \geq f(x)$, (iii) $-f(-1)=f(1)=1$.

Find all such functions $f$.