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.

• 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.