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expain it and give me a answer in english

The function \( f(x, y) \) is defined in a piecewise manner as follows:

\[
f(x, y) = \begin{cases} 
\frac{xy}{x^2 + xy + y^2} & \text{if } (x, y) \neq (0, 0) \\ 
0 & \text{if } (x, y) = (0, 0)
\end{cases}
\]

### Explanation

1. **Piecewise Definition**:
   - For points \((x, y)\) that are **not equal to \((0, 0)\)**, \( f(x, y) \) is given by the expression \(\frac{xy}{x^2 + xy + y^2}\).
   - At the point \((0, 0)\), the function \( f(x, y) \) is defined to be \(0\).

2. **Purpose of the Piecewise Definition**:
   - The function has a specific form for most points in the plane, but at \((0, 0)\), it is assigned a separate value. This kind of piecewise definition is often used when the expression for \( f(x, y) \) is **not defined or may be undefined at \((0, 0)\)**, or to ensure continuity.

3. **Potential Continuity Check**:
   - To determine if \( f(x, y) \) is **continuous at \((0, 0)\)**, we need to check if the limit of \( f(x, y) \) as \((x, y) \to (0, 0)\) matches the defined value at \((0, 0)\), which is \(0\).
   - This involves calculating the **limit**:
     \[
     \lim_{(x, y) \to (0, 0)} \frac{xy}{x^2 + xy + y^2}
     \]
   - If this limit is \(0\), then \( f(x, y) \) is continuous at \((0, 0)\); otherwise, it is not.

4. **Calculating the Limit** (Brief Outline):
   - A common approach is to evaluate the limit along different paths toward \((0, 0)\), such as along \( y = kx \) for some constant \( k \). If the limit value is the same along all paths, then we can conclude the limit exists and equals that value.

Would you like to proceed with calculating the limit to check for continuity at \((0, 0)\)?

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Here are some questions to explore further:
1. What is the limit of \( f(x, y) \) as \((x, y) \to (0, 0)\) along the path \(y = x\)?
2. Does the limit depend on the path taken to approach \((0, 0)\)?
3. What is the significance of defining \( f(0,0) = 0\) separately?
4. How do piecewise-defined functions help in handling undefined points?
5. What mathematical tools are used to determine continuity at a point?

**Tip**: When dealing with piecewise functions, always check for continuity at points where the function definition changes.

Explain the function f(x, y) given in the image and provide an answer in English.

Explore the continuity of the function f(x, y) defined as xy / (x^2 + xy + y^2) for (x, y) ≠ (0,0) and f(x, y) = 0 at (0,0). Learn how to check for continuity by evaluating the limit of f(x, y) as (x, y) approaches (0, 0).

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