Show that if x is a real number, then ⌈x⌉ − ⌊x⌋ = 1 if x is not an integer and ⌈x⌉ − ⌊x⌋ = 0 if x is an integer.

1) $x$ is integer

$\lceil x\rceil -\lfloor x \rfloor =x-x=0$.

2) $x$ is not integer

Let $\lfloor x \rfloor=n$ , then $n<x<n+1$ (or $-n-1<-x<-n$ )

Let $\lceil x \rceil =m$ , then $m-1<x< m$

We have

$(m-1)+(-n-1)< x-x<m-n$

$m-n-2<0<m-n$

$\lceil x\rceil -\lfloor x \rfloor -2<0< \lceil x\rceil -\lfloor x \rfloor$

$0<\lceil x\rceil -\lfloor x \rfloor <2$

Since $\lceil x\rceil -\lfloor x \rfloor$ is integer, it folows that $\lceil x\rceil -\lfloor x \rfloor =1$

So, $\lceil x\rceil -\lfloor x \rfloor =\begin{cases} 0, \quad x\in \mathbb{Z} \\ 1, \quad x\not\in \mathbb{Z} \end{cases}$