As indicated in the main part of this section, functions defined by
piecewise cases are characterized by partitioning the
domain of the function and defining the function value by
identifying in what particular subset of the partition a number
Piecewise Defined Functions and Discontinuity:
Many functions defined by cases have a graphical appearance of a
break or hole in the curve(s). This is described as a
discontinuity or step for the function at the $x$ in the
domain where the break occurs. On a graph such a point is sometimes
visualized by a small circle or a dot. With some technology
the graph may not display the hole or may patch up the step by
connecting steps that have small gaps.
In a mapping diagram the discontinuity can be seen dynamically by a
sudden jump in the behavior of the arrow, a gap where no arrows
occur, or an arrow that seems out of step with arrows from numbers
close in the domain to $x$ where the discontinuity occurs.
We explore how mapping diagrams for
functions defined by cases help us visualize and understand further
the meaning and qualities of these functions.
Though functions defined with piecewise cases often have
discontinuities when the cases change, sometimes there is enough
flexibility in the function to remove the discontinuity by a
choice of a constant, sometimes referred to as a "choice of parameter".
This next example demonstrates that situation and its visualization
with graphs and a mapping diagrams.
Continuous functions defined with cases. [Removing
Example OW.DFDPC.0 :
Visualization of Functions Defined by Piecewise Cases: Graphs and
You supply the functions and the