This demonstration graphs a function f(x,y) given in Cartesian coordinates, which is a monkey saddle by default. The window labeled "Domain: f(x,y)" shows the domain of the grey surface as well as a green disc domain with radius δ.

The center and radius of the green rectangle can be changed by clicking and dragging the red and yellow hotspots respectively. In the graph window, the graph of f(x,y) over this δ domain is shown in green along with two pink plates a distance ε above and below the red point.

To use this demo to test for continuity, start by choosing an ε in the control panel. The challenge then is to see if it is possible to adjust the radius of the green δ domain so that the graph over it lies in between the two plates. If it is always possible to find such a delta for any given epsilon, then the function is continuous at that point.

Make two additional examples visible by clicking either the "Ex1" or "Ex2" checkboxes.  Neither one of these examples are continuous throughout the domain.  Why?