Limit and Continuity of Functions of Two Variables

Limit and Continuity of Functions of Two Variables

Definition 8.6 (Limit of a Function)

Suppose that A {( x , y) | a < x < b , c < y < d } ⊂ ℝ² , F : A → R. We say that F has a limit L at (u,v) if the following hold :

For every neighboourhood ( L − ε , L + ε ), ε > 0 , of L , there exists a δ –neighbourhood B_δ ((u, v )) ⊂ A of (u,v) such that ( x , y) ∈ B_δ ((u, v )) \ {(u, v )}, δ > 0 ⇒ f (x) ∈(L − ε , L + ε ) .

We denote this by lim _{( x , y ) →(u,v)} F ( x, y ) = L if such a limit exists.

When compared to the case of a function of single variable, for a function of two variables, there is a subtle depth in the limiting process. Here the values of F ( x, y) should approach the same value L , as ( x , y) approaches (u , v) along every possible path to (u , v) (including paths that are not straight lines). Fig.8.9 explains the limiting process.

All the rules for limits (limit theorems) for functions of one variable also hold true for functions of several variables.

Now, following the idea of continuity for functions of one variable, we define continuity of a function of two variables.

Definition 8.7 (Continuity)

Suppose that A = {( x, y ) | a < x < b, c < y < d } ⊂ ℝ² , F : A → ℝ. We say that F is continuous at at (u , v) if the following hold :

(1) F is defined at (u , v)

(2) lim_{( x , y ) →(u,v)} F ( x, y ) = L exists

(3) L = F (u,v).

Remark

(1) In Fig. 8.10 taking L = F ( x₀ , y₀ ) will illustrate continuity at ( x₀ , y₀ ) .

(2) Continuity for f (x₁ , x₂ , , x_n ) is also defined similarly as defined above.

Let us consider few examples as illustrations to understand continuity of functions of two variables.

Example 8.8

Let f (x , y) =  (or) 3x−5y+8 / x²+y²+1 for all ( x , y) ∈ ℝ². Show that f is continuous on ℝ².

Solution

Let (a , b) ∈ ℝ² be an arbitrary point. We shall investigate continuity of f at (a , b).

That is, we shall check if all the three conditions for continuity hold for f at (a , b) .

To check first condition, note that f (a, b) =  is defined.

Now we note that lim_{x,y→( a , b)} f (x, y ) = L = f (a, b) . Hence f satisfies all the three conditions for continuity of f at (a , b) . Since (a , b) is an arbitrary point in ℝ² , we conclude that f is continuous at every point of ℝ².

Example 8.9

Consider f ( x, y) =  if ( x , y) ≠ (0, 0) and f (0, 0) = 0 . Show that f is not continuous at (0, 0) and continuous at all other points of ℝ² .

Solution

Note that f is defined for every ( x , y) ∈ R² . First let us check the continuity at ( a , b) ≠ (0, 0) .

Let us say, just for instance, (a , b) = (2, 5) . Then f (2, 5) = 10/29 . Then, as in the above example, we  calculate

it follows that f is continuous at (2, 5) .

Exactly by similar arguments we can show that f is continuous at every point ( a , b) ≠ (0, 0) . Now let us check the continuity at (0, 0) . Note that f (0, 0) = 0 by definition. Next we want to find if

exists or not.

First let us check the limit along the straight lines y = mx , passing through (0, 0).

So for different values of m , we get different values m / 1+ m² and hence we conclude that  does not exist. Hence f cannot be continuous at (0, 0) .

Example 8.10

Consider g ( x , y) =  if ( x , y) ≠ (0, 0) and g(0, 0) = 0. Show that g is continuous on ℝ².

Solution

Observe that the function g is defined for all ( x , y) ∈ ℝ² . It is easy to check, as in the above examples, that g is continuous at all point ( x , y) ≠ (0, 0) . Next we shall check the continuity of g at (0, 0) . For that we see if g has a limit L at (0, 0) and if L = g(0, 0) = 0 . So we consider

Note that in the final step above we have used 2|xy| ≤ x² + y² (which follows by considering 0 ≤ (x − y)² ) for all x , y ∈ ℝ. Note that ( x , y) → (0, 0) implies |x| → 0 . Then from (9) it follows that  = 0 = g(0, 0) ; which proves that g is continuous at (0, 0) . So g is continuous at every point of ℝ².