Q.Evaluate $[3.7]$, $[-2.3]$, $[5]$ and $[-0.6]$, where $[\,\cdot\,]$ denotes the greatest integer function, and state the range of $f(x)=[x]$.

The greatest integer function $[x]$ returns the greatest integer that is less than or equal to $x$.

For a positive non-integer take the integer part; for a negative non-integer step down to the next lower integer, which is the point students most often get wrong.

By definition, $[x]$ is the greatest integer $\le x$ — imagine stepping down the number line from $x$ to the first integer at or below it.

• $[3.7]$: integers $\le 3.7$ are $\dots,2,3$; the greatest is $3$. So $[3.7]=3$.
• $[-2.3]$: integers $\le -2.3$ are $\dots,-4,-3$; the greatest is $-3$ (note $-2$ is not $\le -2.3$). So $[-2.3]=-3$.
• $[5]$: $5$ is itself an integer and $5\le 5$, so $[5]=5$.
• $[-0.6]$: integers $\le -0.6$ are $\dots,-2,-1$; the greatest is $-1$. So $[-0.6]=-1$.

Range. As $x$ ranges over all real numbers, $[x]$ takes every integer value and only integer values, so

Dual check for the negative cases (using $[x]\le x<[x]+1$): for $x=-2.3$, we need $[x]\le -2.3<[x]+1$; $[x]=-3$ gives $-3\le -2.3<-2$. ✓ For $x=-0.6$, $[x]=-1$ gives $-1\le -0.6<0$. ✓

The classic mistake is rounding towards zero for negatives, e.g. writing $[-2.3]=-2$. The greatest integer not exceeding $-2.3$ is $-3$: always step to the integer at or below $x$, which for a negative non-integer is the more negative one.