Section 15.2 Limits and Continuity

Let f be a function of two variables whose domain D includes points arbitrarily close to (a, b). Then we say the limit of f (x, y) as (x, y) approaches (a, b) is L and we write if for every number ε > 0 there is a corresponding δ > 0 such that LIMIT OF A FUNCTION OF TWO VARIABLES

A LIMIT RESULT If f (x, y) → L 1 as (x, y) → (a, b) along a path C 1 and f (x, y) → L 2 as (x, y) → (a, b) along a path C 2, where L 1 ≠ L 2, then

EXAMPLES Graph of Example 2.

CONTINUITY A function f of two variables is called continuous at (a, b) if We say f is continuous on D if f is continuous at every point (a, b) in D.

EXAMPLE The function is continuous except on the parabola y = x 2.

FUNCTIONS OF THREE VARIABLES There are analogous definitions for the limit and continuity for functions of three (or more) variables.