Floor and ceiling functions

• The floor (also called the integral part, integer part, greatest integer, or entier) of a real number $x$ is the greatest integer less than or equal to $x$.
  • (floor function) $\text{floor}(x)=\lfloor x\rfloor :=\max \{m\in \mathbb{Z}\mid m\le x\}$
  • $x-1<\lfloor x\rfloor \le x<\lfloor x\rfloor +1$
  • $n<x<n+1⟹n=\lfloor x\rfloor$ for $n\in \mathbb{N}$
• The ceiling of a real number $x$ is the least integer greater than or equal to $x$.
  • (ceiling function) $\text{ceil}(x)=\lceil x\rceil :=\min \{n\in \mathbb{Z}\mid n\ge x\}$