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Question: []

Indicate whether the following statement is True or False.
\[Let f(x, y) = \frac{x^2 y}{x^3 + y^3} \quad \text{for} \quad (x, y) \neq (0,0),\]
and\[f(0,0) = 0.\]
then f is continuous at (0,0)

To determine whether the function \( f(x, y) = \frac{x^2 y}{x^3 + y^3} \) is continuous at the point \((0,0)\), we must rigorously analyze the behavior of \( f(x, y) \) as \((x, y)\) approaches \((0,0)\). The function is defined as:

\[
f(x, y) = \frac{x^2 y}{x^3 + y^3} \quad \text{for} \quad (x, y) \neq (0,0),
\]
with
\[
f(0,0) = 0.
\]

### Step 1: Definition of Continuity
A function \( f(x, y) \) is continuous at a point \((a, b)\) if:
\[
\lim_{(x, y) \to (a, b)} f(x, y) = f(a, b).
\]
In this case, we need to evaluate:
\[
\lim_{(x, y) \to (0,0)} \frac{x^2 y}{x^3 + y^3}
\]
and check whether it equals \(f(0,0) = 0\).

### Step 2: Approach Along Different Paths
We will approach \((0,0)\) along different paths to see if the limit is consistent:

**Path 1: \( y = 0 \):**
\[
f(x, 0) = \frac{x^2 \cdot 0}{x^3 + 0^3} = 0 \quad \text{for} \quad x \neq 0.
\]
So, \(\lim_{x \to 0} f(x, 0) = 0\).

**Path 2: \( x = 0 \):**
\[
f(0, y) = \frac{0^2 \cdot y}{0^3 + y^3} = 0 \quad \text{for} \quad y \neq 0.
\]
Thus, \(\lim_{y \to 0} f(0, y) = 0\).

**Path 3: \( y = x \):**
\[
f(x, x) = \frac{x^2 \cdot x}{x^3 + x^3} = \frac{x^3}{2x^3} = \frac{1}{2} \quad \text{for} \quad x \neq 0.
\]
Here, \(\lim_{x \to 0} f(x, x) = \frac{1}{2}\).

### Step 3: Conclusion
Since the limit depends on the path chosen, and in particular, \(\lim_{(x, y) \to (0,0)} f(x, y)\) is not the same along different paths, the limit does not exist.

Thus, \( \lim_{(x, y) \to (0,0)} f(x, y) \) does **not** equal \(f(0,0) = 0\). Consequently, \(f(x, y)\) is **not** continuous at \((0,0)\).

### Final Answer:
The statement is **False**.

---

Do you want further details or have any questions?

Here are 8 related questions for further exploration:
1. What are the general conditions for the continuity of a multivariable function?
2. How does one formally define a limit in two variables?
3. What are common strategies for evaluating limits of functions of two variables?
4. Can a function be discontinuous at a point even if it’s continuous along every path to that point?
5. How does path dependence affect the continuity of multivariable functions?
6. What role does polar coordinates play in evaluating limits of multivariable functions?
7. How can the concept of epsilon-delta be extended to functions of two variables?
8. What are examples of functions that are continuous everywhere except at one point?

**Tip:** When evaluating limits for multivariable functions, always check for consistency across different paths to ensure the limit is well-defined.

Explore the continuity analysis of a multivariable function f(x, y) = (x^2 * y) / (x^3 + y^3) at the point (0,0). Learn how to evaluate limits in two variables and understand the conditions for continuity. Suitable for college-level students and anyone interested in advanced calculus topics.

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