Step Functions

# Step Functions

We will soon look at Riemann-Stieltjes integrals where the integrator $\alpha$ is a step function, however, we will first need to formally define what exactly a step function is.

 Definition: A Step Function $\alpha$ on the interval $[a, b]$ is a piecewise constant function containing finitely many pieces, i.e., there exists a partition $P = \{a = x_0, x_1, ..., x_n = b \} \in \mathscr{P}[a, b]$ such that $\alpha (x)$ constant for all $x \in (x_{k-1}, x_k)$ for each $k \in \{1, 2, ..., n \}$. The Jump at $x_k$ for $k \in \{0, 1, 2, ..., n \}$ is defined to be $\alpha(x_k^+) - \alpha(x_k^-)$. For $k = 0$ the jump at $x_0$ is defined to be $\alpha(x_0^+) - \alpha(x_0)$, and for $k = n$, the jump at $x_n$ is defined to be $\alpha (x_n) - \alpha(x_n^-)$.

For example, consider the function $\alpha$ defined on the interval $[0, 3]$ by:

(1)
\begin{align} \quad \alpha (x) = \left\{\begin{matrix} 1 & \mathrm{if} \: 0 \leq x \leq 1 \\ 2 & \mathrm{if} \: 1 < x \leq \frac{3}{2}\\ 4 & \mathrm{if} \: \frac{3}{2} < x \leq 2\\ -1 & \mathrm{if} \: 2 < x \leq 3 \end{matrix}\right. \end{align}

Then $\alpha$ is indeed a step function because $\alpha (x)$ is constant on the intervals $(0, 1)$, $\left ( 1, \frac{3}{2} \right )$, $\left ( \frac{3}{2}, 2 \right )$ and $(2, 3)$ corresponding to the partition $P = \left \{ 0, 1, \frac{3}{2}, 2, 3 \right \} \in \mathscr{P}[0, 3]$. The graph of $\alpha$ is given below: Notice that the points of discontinuities of step functions are the "joining" points of these subintervals. In the example above, we see that locations of possibly discontinuities are $x_0 = 0$, $x_1 = 1$, $x_2 = \frac{3}{2}$, $x_3 = 2$, and $x_4 = 3$.

It is important to note that given an arbitrary partition $P = \{ a = x_0, x_1, ..., x_n = b \} \in \mathscr{P} [a, b]$, by the definition of a step function that if $\alpha$ is a step function that is constant on each open subinterval $(x_{k-1}, x_k)$ for each $k \in \{1, 2, ..., n \}$ then $\alpha$ need not be left or right continuous at each of the points $x_0, x_1, x_2, ..., x_n$.