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.

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 $r$.