Discontinuous Surface

Discontinuous Surface

12.3 Limits and continuity

Note: Section numbers refers to Calculus, Early Transcendentals (2nd edition) by Briggs, Cochran, and Gillett.

A function of two (or more) variables is continuous at a point, provided its limit equals its value at that point.
The function

is discontinuous at (x,y)=(0,0). 

The graph is a surface of fixed height along each line through the origin - a very hard concept to wrap the mind around. This creates the fold in the middle. Note in the print below that there are four fixed height lines (in purple) to emphasize this strange fact. 

The title of this post reflects the fact that the graph looks very much like a specialty pasta (See http://recipes.wikia.com/wiki/File:Funghini.jpg).