Step Functions

Step Function: Comprehensive Guide
1. Fundamental Concepts
Definition: A step function is a piecewise constant function. Its graph looks like a series of “steps,” since the value remains constant within each interval and jumps at certain points.

Notation:
The most common step function is the floor function ( $\lfloor x \rfloor$ ) or greatest integer function. However, more general step functions may be defined on intervals with different constant values.
Graphical Representation:
Horizontal line segments represent constant values on each interval.
Jumps (discontinuities) occur where the function switches to the next constant value.

2. Key Concepts
General Rule: A step function can be written in the form
$$f(x) = c_i \quad \text{for } x \in I_i$$ where each $I_i$ is an interval and $c_i$ is a constant. Special Case (Floor Function): $$f(x) = \lfloor x \rfloor = n \quad \text{if } n \leq x < n+1$$

Applications:
Used in computer science (rounding down values).
In probability/statistics to define piecewise distributions.
In economics to model sudden price jumps or thresholds.

3. Examples
Example 1 (Basic Step Function):
$$f(x) = \begin{cases} 1 & 0 \leq x < 2 \\3 & 2 \leq x < 4 \\5 & 4 \leq x < 6\end{cases}$$
Graph: three horizontal steps at heights 1, 3, and 5.
Example 2 (Greatest Integer Function):
Evaluate $\lfloor 2.7 \rfloor$ .
The greatest integer $\leq 2.7$ is 2. So, $\lfloor 2.7 \rfloor = 2$ .

4. Problem-Solving Techniques
Number Line Method: Plot the value of $x$ on a number line, then drop down to the constant assigned in that interval.
Check Endpoints: Always verify which interval includes the given $x$ .
Graph Reading: Practice sketching graphs to visualize how the function jumps.