Question 1
Find all functions $f:\mathbb R\to\mathbb R$ such that:
(i) $f(x)+f(y)+1\ge f(x+y)\ge f(x)+f(y)$ for all real $x,y$;
(ii) $f(0)\ge f(x)$ for all $x\in[0,1)$;
(iii) $-f(-1)=f(1)=1$.
(i) $f(x)+f(y)+1\ge f(x+y)\ge f(x)+f(y)$ for all real $x,y$;
(ii) $f(0)\ge f(x)$ for all $x\in[0,1)$;
(iii) $-f(-1)=f(1)=1$.