Greatest Integer Function (Step Function)

3. Greatest integer function (step function). Definition: for every real $x$, $f(x)=[x]=$ the greatest integer less than or equal to $x$. $[x]$ is also called the floor function, sometimes written $\lfloor x\rfloor$.

Illustrations: (1) $f(5.7)=[5.7]=$ the greatest integer $\le5.7$. Integers $\le5.7$ are $5,4,3,2,\ldots$, of which $5$ is the greatest, so $[5.7]=5$. (2) $f(-6.3)=[-6.3]=$ the greatest integer $\le-6.3$. Integers $\le-6.3$ are $-10,-9,-8,-7,\ldots$, of which $-7$ is the greatest, so $[-6.3]=-7$. (3) $f(2)=[2]=2$ (an integer's floor is itself). (4) $[\pi]=3$. (5) $[e]=2$.

The function can be defined piecewise as $f(x)=n$ if $n\le x<n+1$, i.e. $x\in[n,n+1)$, $n\in I$ (Fig. 6.45). Domain: $R$; Range: $I$ (the set of integers). …