Use the Sage or the GeoGebra figures and table below to experiment further with functions defined by piecewise cases.

Martin Flashman, Dec 20, 2013, Created with GeoGebra

Notice how the points on the graph are paired with the points on the mapping diagram.Given a point / number, $x$, on the source line, there is a unique arrow meeting the target line at the point / number, $f(x)$. which corresponds to the function's value for $x$

Check the box in GeoGebra to see more points and arrows corresponding to data on the table.

You can change the function and the values of $a$ and $b$ where the "steps" occur to explore this further.