记$x=n+r$和$y=m+s$,其中$n,m\in\Bbb Z$和$r,s\in\left[0,1\right),r\ge s$.
则$f(x,y)=f(r,s)$,$g(x,y)=g(r,s)=[2x]+[2y]-[x+y]$.
Case 1. $r,s\in\left[0,0.5\right)$.
$\{r+0.5\}+\{s+0.5\}=r+s+1\in[1,2)⇒f(r,s)=0$.
$[2r]=[2s]=[r+s]=0⇒g(r,s)=0$.
Case 2. $r\in[0.5,1),s\in[0,0.5)$.
$\{r+0.5\}+\{s+0.5\}=r+s\in[0.5,1.5)⇒f(r,s)=1-[r+s]$.
$[2r]=1,[2s]=0⇒g(r,s)=1-[r+s]$.
Case 3. $r,s\in[0.5,1)$.
$\{r+0.5\}+\{s+0.5\}=r+s-1⇒f(r,s)=2-[r+s]$.
$[2r]=[2s]=1⇒g(r,s)=2-[r+s]$. |
