Continuity of a function of two variables

 Goal: State the conditions for continuity of a function of two variables

Definition

The definition of the continuity of a function of two variables is similar to that of a function of one variable.. 

A function f(x, y) is continuous at a point (a,b) in its domain if the following conditions are verified:

Example 1

Show that the function:

                                                                

Solution
In order for f(a,b) to exist the denominator must be different of zero. We have x + y + 1 = 5 - 3 + 1 =3. Since the denominator is different of zero, let's calculate f(a, b):

Let's calculate limf(x, y) when (x,y) approaches (5, -3):

We have :

All three condiitions are verified. Therefore the function is continuous at the point (5, -3).

Practice

Show that the function: