This applet is designed to help students
develop a visual intuition about the meaning of continuity for
functions of two variables.
Recall that for a function f of one
variable x, the function is continuous at x=x0, if for any
positive ε, we
can
find a positive δ such that a box centered at (x0,f(x0)) going
up and down ε and right and left δ will trap the
function so it goes out of the sides rather than the top and bottom.
While proving continuity requires us
to produce a rule for all positive ε, we often will be convinced
if we can find an ε when δ has values .1, .05, and .01.
For functions of two variable the
definition changes only slightly. The 2-D box becomes a round
"hat box" with radius ε. Once again, we will be pretty
convinced if when δ is .1, .05, and .01, we can find an ε
with the surface only exiting the sides of the box, missing the top and
bottom.
Click on the button below to launch the
applet.
The applet has a control window and 2 graphics windows.
The values of the coordinates are on the control panel. The
values of x0, y0, ε, and δ can either be typed in or shifted with the
arrow buttons.
The graph window shows a graph
of the function, with a red dot at (x0, f0, f(x0,y0)) and a green patch
for the graph of the green
circle of radius ε. It also has two disks of radius ε
at a height of δ above and below the graph of the hot spot. The
hat bow is completed with a light blue side surface.
The close up window zooms in on the disks. For a given
ε, you want to find a δ where the green surface is trapped
between the two pink surfaces, with the gray extension leaving the box
through the sides rather than the top or bottom.
Things to notice:
Sometimes it is hard to understand continuity since almost all
functions we can think of are continuous almost everywhere.
Consider the function f(x,y) = atan(x/y)*sqrt(x^2+y^2). It is
continuous everywhere except at the origin. Compare what happens
at the origin with what at the origin with what happens a small
distance from the origin.