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 lies.

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.
Example OW.FDPC.1 : A function with a single exception in its definition.
Example OW.FDPC.2 : A function with two cases on intervals. [Two rates.]
Example OW.FDPC.3 : A function with three cases on intervals [Three rates.]

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.

Example OW.FDPC.4 : Continuous functions defined with cases. [Removing discontinuities.]

Example OW.DFDPC.0 :Dynamic Visualization of Functions Defined by Piecewise Cases: Graphs and Mapping Diagrams.
You supply the functions and the "cuts."