A function z = f(x,y) is continuous at (a,b) if:

1) The limit as (x,y) approaches (a,b) of f(x,y) exists.
2) f(a,b) exists, and
3) The value of the limit in 1) is the function value f(a,b)

These 3 conditions are often compressed into a single statement:
If the limit as (x,y) approaches (a,b) of f(x,y) is f(a,b), then f(x,y) is continuous at (a,b).

All of the examples on the tangent plane page are continuous everywhere. To see examples of discontinuities, check out the limits page. The only function there which is continuos at (0,0) is the first. Can you tell which property the other fail to satisfy at (0,0)?

Back to the tangent plane page.