Examine the following function for continuity at the point (0, 0), where f(0, 0) = 0 and f(x, y) for (x, y) != (0, 0) is given by |x| + |y|

To examine the continuity of the function \( f(x, y) \) at the point \((0, 0)\), we need to verify if the limit of \( f(x, y) \) as \((x, y)\) approaches \((0, 0)\) equals \( f(0, 0) \), which is given as 0. Specifically, we check:

\[
\lim_{(x, y) \to (0, 0)} f(x, y) = f(0, 0)
\]

### The function is defined as:
\[
f(x, y) =
\begin{cases}
|x| + |y|, & (x, y) \neq (0, 0) \\
0, & (x, y) = (0, 0)
\end{cases}
\]

For the function to be continuous at \((0, 0)\), the limit as \((x, y)\) approaches \((0, 0)\) must be equal to 0. Let's examine this limit.

### Approach along different paths:
1. **Along the x-axis** \((y = 0)\):
   \[
   f(x, 0) = |x| + |0| = |x|
   \]
   \[
   \lim_{x \to 0} f(x, 0) = \lim_{x \to 0} |x| = 0
   \]

2. **Along the y-axis** \((x = 0)\):
   \[
   f(0, y) = |0| + |y| = |y|
   \]
   \[
   \lim_{y \to 0} f(0, y) = \lim_{y \to 0} |y| = 0
   \]

3. **Along the line \( y = x \)**:
   \[
   f(x, x) = |x| + |x| = 2|x|
   \]
   \[
   \lim_{x \to 0} f(x, x) = \lim_{x \to 0} 2|x| = 0
   \]

4. **Along the line \( y = -x \)**:
   \[
   f(x, -x) = |x| + |-x| = 2|x|
   \]
   \[
   \lim_{x \to 0} f(x, -x) = \lim_{x \to 0} 2|x| = 0
   \]

### General approach along any path:
Consider the general path \( y = mx \) (where \( m \) is a constant):
\[
f(x, mx) = |x| + |mx| = |x|(1 + |m|)
\]
\[
\lim_{x \to 0} f(x, mx) = \lim_{x \to 0} |x|(1 + |m|) = 0
\]

Thus, in all these cases, the limit as \((x, y)\) approaches \((0, 0)\) is 0.

### Conclusion:
Since the limit of \( f(x, y) \) as \((x, y) \to (0, 0)\) is 0 and \( f(0, 0) = 0 \), the function \( f(x, y) \) is **continuous at the point \((0, 0)\)**.

Would you like further details or have any questions?

### 5 Related Questions:
1. What is the definition of continuity for multivariable functions?
2. How does the method of approaching a point along different paths help in determining continuity?
3. Can the absolute value function introduce discontinuities in other contexts?
4. What happens if the limit along different paths yields different values?
5. How can polar coordinates be used to analyze limits in two dimensions?

### Tip:
Always test continuity from multiple directions or paths when dealing with functions of several variables to ensure that the limit is consistent.

Examine the following function for continuity at the point (0, 0), where f(0, 0) = 0 and f(x, y) for (x, y) != (0, 0) is given by |x| + |y|.

Analyze the continuity of the multivariable function f(x, y) = |x| + |y| at the point (0, 0). Learn how to apply limit-based methods and explore different paths to determine if the function is continuous. This solution is ideal for students in Grades 11-12 and early university calculus courses.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation