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To determine the continuity of the given function \( f(x, y) \), we need to check if the function is continuous at all points in its domain, including the origin \((0, 0)\).

### Steps to Determine Continuity:
1. **Away from \((0, 0)\):**
   The function \( f(x, y) \) is defined as
   \[
   f(x, y) = \frac{x^4 - x^3y^2}{x^2 + y^2}, \quad \text{when } x^2 + y^2 \neq 0.
   \]
   For \( x^2 + y^2 \neq 0 \), the denominator is non-zero, and the function is well-defined. Since it is a rational function and continuous wherever the denominator is non-zero, \( f(x, y) \) is continuous for all points where \( x^2 + y^2 \neq 0 \).

2. **At the Origin \((0, 0)\):**
   At \((0, 0)\), the function is defined as \( f(0, 0) = 1 \). To check continuity at this point, we calculate the limit of \( f(x, y) \) as \((x, y) \to (0, 0)\) and compare it with \( f(0, 0) = 1 \).

   We analyze the limit:
   \[
   \lim_{(x, y) \to (0, 0)} f(x, y) = \lim_{(x, y) \to (0, 0)} \frac{x^4 - x^3y^2}{x^2 + y^2}.
   \]

   - Using **polar coordinates**: Let \( x = r\cos\theta \) and \( y = r\sin\theta \), where \( r = \sqrt{x^2 + y^2} \). Substituting:
     \[
     f(x, y) = \frac{(r\cos\theta)^4 - (r\cos\theta)^3(r\sin\theta)^2}{r^2}.
     \]
     Simplify the numerator:
     \[
     f(x, y) = \frac{r^4 \cos^4\theta - r^5 \cos^3\theta \sin^2\theta}{r^2} = r^2 (\cos^4\theta - r\cos^3\theta \sin^2\theta).
     \]

     As \( r \to 0 \), the term \( r^2 (\cos^4\theta - r\cos^3\theta \sin^2\theta) \to 0 \). Thus:
     \[
     \lim_{(x, y) \to (0, 0)} f(x, y) = 0.
     \]

   Since the limit \( \lim_{(x, y) \to (0, 0)} f(x, y) = 0 \) does not equal \( f(0, 0) = 1 \), the function \( f(x, y) \) is **not continuous at \((0, 0)\)**.

### Conclusion:
The function \( f(x, y) \) is continuous everywhere except at the origin \((0, 0)\), where it is discontinuous.

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Do you have any further questions, or would you like to delve into more detailed aspects of this result?

### Questions for Exploration:
1. Why does the polar coordinate substitution simplify the limit calculation?
2. How do we confirm the continuity for \( x^2 + y^2 \neq 0 \)?
3. What role does the definition of \( f(x, y) \) at \((0, 0)\) play in determining continuity?
4. Could modifying \( f(0, 0) \) make the function continuous everywhere?
5. What are other approaches to test continuity in two-variable functions?

### Tip:
When analyzing continuity of multivariable functions at a point, polar coordinates can help simplify cases with radial symmetry or where terms depend on \( r \).

Explore the continuity of the function f(x, y) given by f(x, y) = (x^4 - x^3y^2) / (x^2 + y^2) for x^2 + y^2 ≠ 0, and f(0, 0) = 1. Learn the detailed step-by-step process of checking continuity, using limits and polar coordinates. Suitable for undergraduate calculus students.

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