The Greatest Integer Function

# The Greatest Integer Function

Recall from the Step Functions page that a step function $\alpha$ is a function on the interval $[a, b]$ is a piecewise constant function containing finitely many pieces. In other words, there exists a partition $P = \{a = x_0, x_1, ..., x_n = b \} \in \mathscr{P}[a, b]$ such that $\alpha (x)$ is constant on each subinterval $(x_{k-1}, x_k)$.

Furthermore, jump discontinuities of $\alpha$ occur at the points $x_1, x_2, ..., x_n$.

We will now define a very important step function known as the greatest integer function.

 Definition: The Greatest Integer Function takes every $x \in \mathbb{R}$ and maps it to the greatest integer less than or equal to $x$ and is denoted $[x]$. For all $x \in \mathbb{R}$, $[x] \leq x < x + 1$.

The graph of the greatest integer function is given below:

For example, $[2.4] = 2$. $[\pi] = 3$. $[5.999] = 5$, and $[-4.2] = - 5$.

Furthermore, if we define $\alpha$ to be the step function defined as the greatest function on the interval $[0, 5]$ then the jump discontinuities of occur at $x_1 = 1$, $x_2 = 2$, $x_3 = 3$, $x_4 = 4$, and $x_5 = 5$.