Continuity Applet

This applet is designed to help students develop a visual intuition about the meaning of continuity for functions of one variable.

• 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, .01, and .001.
Click on the button below to launch the applet.
• The applet has a control window and 2 graphics windows.
• The graph window shows a graph of the function, with a red dot at the graph at x0 and a magenta patch on the axis for the epsilon neighborhood of x0 and a corresponding magenta patch on the graph.  It also has a light blue box of  radius δ at a height of ε above and below the graph of x0.
• The values of the coordinates are on the control panel.  The values of ε , δ , and x0 can either be typed in or shifted with the arrow buttons.
• The close up window zooms in on the disks.  For a given ε, you want to find a δ where the magenta curve is trapped in the blue box.
• The tools menu on the graphics windows lets you change the action of a mouse drag between rotations and translations of the image.

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) = sign(x).  It is continuous everywhere except at the origin.  Compare what happens at the origin with what happens a small distance from the origin.
• When we zoom in on any reasonable function, it eventually approximates a line.  For a line delta works as long as epsilon/delta is greater than the slope of the line.  How does this help you find a general rule for sufficiently small epsilon with nice functions?
A Continuity Applet worksheet is available.

The idea of continuity in one variable can be generalized to understand continuity in 2 variables.

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.