In this demo, we show slice of the graph of f(x,y,z). You can choose which slice is shown by rotating the normal vector to the slice plane in the "Plane Orientation" window.
You can also choose the point P at which the continuity is tested by changing the expression for P in the control window.
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 δ so that the graph of the slice lies between the two plates. If the function is continuous at point P then for every possible orientation of the slice plane there must be some δ small enough that the graph of the slice lies between the two plates.