Examine the continuity of following function at (1, 1)
f(x, y) :=
(
1 if x + y ≥ 2,
−1 if x + y < 2.

To examine the continuity of the function \( f(x, y) \) at the point \( (1, 1) \), we must check if the following condition holds:

\[
\lim_{(x, y) \to (1, 1)} f(x, y) = f(1, 1)
\]

### 1. Define the function:
The function is defined as:

\[
f(x, y) = 
\begin{cases}
1 & \text{if } x + y \geq 2, \\
-1 & \text{if } x + y < 2.
\end{cases}
\]

### 2. Evaluate the function at \( (1, 1) \):

At the point \( (1, 1) \), we have:

\[
x + y = 1 + 1 = 2.
\]

Since \( x + y \geq 2 \), we use the first case of the function definition. Thus,

\[
f(1, 1) = 1.
\]

### 3. Compute the limit as \( (x, y) \to (1, 1) \):

We now check the limit of the function as \( (x, y) \) approaches \( (1, 1) \). This involves approaching \( (1, 1) \) from different directions and checking whether the limit is the same.

#### Case 1: Approaching along a path where \( x + y \geq 2 \)
If we approach \( (1, 1) \) along a path where \( x + y \geq 2 \), such as along the line \( y = 2 - x \) for \( x \geq 1 \), the value of the function is:

\[
f(x, y) = 1.
\]

Hence, the limit along this path is:

\[
\lim_{(x, y) \to (1, 1)} f(x, y) = 1.
\]

#### Case 2: Approaching along a path where \( x + y < 2 \)
If we approach \( (1, 1) \) along a path where \( x + y < 2 \), such as along the line \( y = 2 - x \) for \( x < 1 \), the value of the function is:

\[
f(x, y) = -1.
\]

Hence, the limit along this path is:

\[
\lim_{(x, y) \to (1, 1)} f(x, y) = -1.
\]

### 4. Conclusion:
The limits from different directions do not agree:

\[
\lim_{(x, y) \to (1, 1)} f(x, y) \neq 1.
\]

Thus, the function does not have a limit as \( (x, y) \to (1, 1) \). Since the limit does not exist, the function is **not continuous** at \( (1, 1) \).

---

Would you like more details on any part of this? Here are 5 related questions to explore further:

1. What are the conditions for a function to be continuous at a point?
2. How can we rigorously compute limits in two variables?
3. What does the difference between one-sided limits tell us about continuity?
4. How do discontinuities impact the behavior of a function near a point?
5. Can a function be continuous along some paths but not others?

**Tip:** Always check limits from different directions when dealing with multivariable functions to ensure continuity.

This page examines the continuity of a piecewise function f(x, y) at the point (1, 1). Learn how to evaluate the limit from different directions and apply continuity theorems to determine if the function is continuous. Suitable for students studying Multivariable Calculus.

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