According to the epsilon-delta definition, a function f of three real variables is said to be continuous at (x0,y0,z0)) if for any ε > 0 there exists a &delta such that | f(x,y,z) - f(x0,y0,z0) | < ε whenever | (x,y,z) - (x0,y0,z0) | < &delta.
In this demo, we show a z-slice of the graph of f(x,y,z): We fix the variable z=z0; the value of z0 can be changed using the tapedeck controllers in the control panel. The window labeled "Domain: f(x,y,z) 3D Picture " shows the domain of the function f(x,y,z) as well as a green cubic domain around the (red) hotspot P.
You can move P around by grabbing and dragging it, and note that you can change the sidelength &delta of the domain by moving around the yellow hotspot in the 2D lattice of the "Domain: f(x,y,z) 2D Picture" window. The bar next to the lattice shows the height of the green domain and is determined by and, of course, equal to the value of &delta.
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 sidelength of the green &delta domain so that the graph over it lies in between the two square plates. Note that because the function f depends on three variables, we cannot take a look at the graph of the function to determine whether the &delta we have found for the given ε works. However, what we can do is to look at the slices, and see whether we have found a &delta that works for each of the slices.
Now you have to find a &delta so that the slice f(x,y,z0) always lies between the two plates for all values of z0. If (and only if) it is possible to find such a delta for any given epsilon, then the function is continuous at point P.
ExercisesFor the function f(x,y,z)= x2+y2-z2, set ε to 0.2, 0.1 and 0.05, respectively. Then for each of these values of ε, use the tapedeck controller to find a corresponding &delta.