Greatest Integer Function

The greatest integer function, often written as $f(x) = [x]$ or $\lfloor x \rfloor$, is a special kind of step function that rounds any real number down to the nearest integer less than or equal to it. For example, $[3.2] = 3$, $[-1.5] = -2$, and $[4] = 4$. Its graph looks like a staircase, with a sudden vertical jump at every integer — this jump is what makes the function discontinuous at every integer point. Understanding this behaviour is essential because it introduces the idea of a f …