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.