To analyze the continuity of the function:

$f(x, y) = \frac{xy}{\sqrt{x^4 + y^4}}$

at the point $(0, 0)$, we need to check if the limit exists and if the function can be defined in such a way that it is continuous at $(0, 0)$.

1. Check if the limit exists at $(0, 0)$

We examine the limit of the function as $(x, y) \to (0, 0)$. To do this, we can approach the origin along different paths and see if the limit is the same along all these paths.

Path 1: Along the line $y = 0$

Substitute $y = 0$ into the function: $f(x, 0) = \frac{x(0)}{\sqrt{x^4 + 0^4}} = 0$ So, $\lim_{x \to 0} f(x, 0) = 0$.

Path 2: Along the line $x = 0$

Substitute $x = 0$ into the function: $f(0, y) = \frac{(0)y}{\sqrt{0^4 + y^4}} = 0$ So, $\lim_{y \to 0} f(0, y) = 0$.

Path 3: Along the line $y = x$

Substitute $y = x$ into the function: $f(x, x) = \frac{x(x)}{\sqrt{x^4 + x^4}} = \frac{x^2}{\sqrt{2x^4}} = \frac{x^2}{\sqrt{2}x^2} = \frac{1}{\sqrt{2}}$ The limit along this path is $\frac{1}{\sqrt{2}}$, which is not equal to zero.

2. Conclusion

Since the limit depends on the path taken (i.e., along the lines $y = 0$ and $x = 0$, the limit is 0, but along the line $y = x$, the limit is $\frac{1}{\sqrt{2}}$), the limit does not exist as $(x, y) \to (0, 0)$.

Therefore, the function is not continuous at $(0, 0)$ because the limit does not exist at that point.

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Let me know if you would like more details on how to define the function or analyze the behavior. Here are five relative questions to expand:

1. How can we formally define continuity at a point for a function of two variables?
2. How would the result change if we approached $(0, 0)$ along other curves like $y = x^2$?
3. What is the importance of checking different paths when analyzing limits in two variables?
4. How does this example illustrate the difference between single-variable and multivariable continuity?
5. Could we redefine the function at $(0, 0)$ to make it continuous?

Tip:

When analyzing the continuity of multivariable functions, always check the limit along different paths to confirm if it depends on the direction of approach.