To access the multi-variable continuity applet (based
on work of Tom Banchoff) follow the link:
(Before doing this worksheet, it would be wise to do the worksheet on continuity in 1 variable. That worksheet is at:
Spread the three graphs out
where you can see them all, as well as the control panel.
The applet opens with the graph of the function
= .5*(x^2-y^2)/(x^2+y^2), a rational function which is obviously continuous except at the origin.
(This function is easier to understand in polar coordinates where it is cos(2θ)/2.)
We will start at an easy point and me to harder points. The applet has an f(x,y) graph window, a
close up graph window, and a domain window. Visually the
graph of the function in an δ-neighborhood of (x0, y0, f(x0, y0)) is
a green patch. Two pink lines
are drawn ε above and below the point. The definition
of continuity says that no matter how small a value of ε we are given, δ can be
chosen so that the green patch is trapped by the pink disks.
example starts with f(x,y)
= .5*(x^2-y^2)/(x^2+y^2), x0 = y0 = 0.4, and ε
= δ = 0.1. Find a value of δ that works
with this ε at
this point. (Make δ smaller until the green patch is
between between the two disks.)
Now reduce ε to 0.05 and find a δ that works for that ε and that point.
Repeat the process with ε = 0.01.
Notice that we have zoomed in far enough for the green patch to look like a piece of the plane. Conjecture a rule that will produce a good δ for any ε < 0.01. (Find a ratio of δ/ ε that always produces a good δ if ε is small enough for the graph to look linear.
2) Repeat the process above with a new point. Set x0=0.2, y0=0.1. Find values of δ that work at this point when ε = 0.1, 0.05, and 0.01. Conjecture a rule that will produce a good δ for any ε < 0.01.
3) The applet won't let us examine the function at the origin
directly since the function is not defined there. Instead we will
look at a sequnce of settings to conclude the divisior needed to turn
an epsilon into a good delta grows without bound as you approach the
On the Close-up window, change the view to "Up Z Axis". If delta is small enough we will see only pink. If delta is too big we will be able to see green. We will look at the points (x0, y0) = (10^-n, 10^-n), where n = 2, 3, 4, ...
Set ε = δ = 0.1. For x0 and y0 enter 1E-2. You should see a lot of green. Change δ to 1E-2 and note that that is not small enough. Change δ to 1E-3 to see that that is small enough. Change δ to 2E-3, which is not small enough to be a good delta, but is close enough to see that a good delta exists.
Repeat the process with x0 = y0 = 1E-3 and δ varying between 1E-3 and 1E-4.
Repeat the process with x0 = y0 = 1E-5 and δ varying between 1E-5 and 1E-6.
Explain your conclusions about continuity of f(x,y) along the line x=y.
4) Discuss the continuity of the function f(x,y) = sign(x*y).
5) Discuss the continuity of f(x,y) = sin(x^2+y^2)/(x^2+y^2).
Be sure to address continuity at the origin. Can the
function be defined at the origin in a way that makes the function
Return to the Multi-variable Continuity Applet page.
Last updated By Mike May, S.J., January 25, 2006.