To access the multi-variable continuity applet (based
on work of Tom Banchoff) follow the link:

http://www.slu.edu/classes/maymk/banchoff/Continuity.html

(Before doing this worksheet, it would be wise to do the worksheet on continuity in 1 variable. That worksheet is at:

http://www.slu.edu/classes/maymk/banchoff/Continuity1VarWS.html)

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
f(x,y)
= .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.

1) The
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
origin.

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
continuous?

Return to the Multi-variable Continuity Applet page.

Last updated By Mike May, S.J.,
January 25, 2006.