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.

Many functions defined by cases have a graphical appearance of a break or hole in the curve(s). This is described as

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.You supply the functions and the "cuts."