Continuity Applet

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.
A Multi-variable Continuity Applet worksheet is available.

This applet was designed as by modifying an applet from a demo by Tom Banchoff at Brown University.  It is used with permission.   Go to the Banchoff Applet Help page.

Return to the Banchoff Applet page.
Return to the SLU Calculus Applet page.

Return to the Saint Louis University Department of Mathematics and Computer Science home page

Last updated By Mike May, S.J. ,  October 6, 2006.