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To determine the set of all non-negative values of \( k \) for which the function \( f \) is continuous at \( (0,0) \), we need to analyze the given function:

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

### Continuity at \( (0,0) \)

For \( f \) to be continuous at \( (0,0) \), the limit of \( f(x,y) \) as \( (x,y) \) approaches \( (0,0) \) must equal \( f(0,0) \), which is 0. 

So, we need to check:

\[
\lim_{(x,y) \to (0,0)} \frac{2xy}{(x^2 + y^2)^k} = 0
\]

### Polar Coordinates

It is useful to convert the function into polar coordinates:

\[
x = r \cos \theta, \quad y = r \sin \theta
\]

Then,

\[
f(x,y) = \frac{2xy}{(x^2 + y^2)^k} = \frac{2(r \cos \theta)(r \sin \theta)}{(r^2)^k} = \frac{2r^2 \cos \theta \sin \theta}{r^{2k}} = 2 \cos \theta \sin \theta \cdot r^{2-2k}
\]

### Limit Analysis

To find the limit as \( r \to 0 \):

\[
\lim_{r \to 0} 2 \cos \theta \sin \theta \cdot r^{2-2k}
\]

For the limit to be zero, the exponent \( 2-2k \) must be positive (since \( \cos \theta \sin \theta \) is bounded). Therefore:

\[
2 - 2k > 0 \implies 2 > 2k \implies k < 1
\]

### Conclusion

The function \( f \) is continuous at \( (0,0) \) if \( k < 1 \).

Thus, the set of all non-negative values of \( k \) for which \( f \) is continuous at \( (0,0) \) is:

\[
\boxed{[0,1)}
\]

Hence, the correct answer is:

(B) \( \{ k \in \mathbb{R} : 0 \leq k < 1 \} \)

Do you want any more details or have any questions?

Here are 8 relative questions that expand on this problem:

1. How would the continuity condition change if \( k \) could also be negative?
2. What is the behavior of the function as \( k \) approaches 1 from the left?
3. How do different values of \( \theta \) affect the limit analysis in polar coordinates?
4. Can you explain the significance of converting to polar coordinates in continuity problems?
5. What would the function look like for specific values of \( k \) within the range \( [0,1) \)?
6. How would you prove the continuity for a specific value of \( k \) in the allowed range?
7. What happens to the function if \( k = 1 \)?
8. How does the choice of the definition of \( f \) at \( (0,0) \) affect its continuity?

**Tip:** When analyzing the continuity of functions at a point, always consider converting to polar coordinates if the function involves \( x^2 + y^2 \). This can simplify the problem significantly.

Explore the continuity of a function f(x,y) at the point (0,0) by analyzing limits and converting to polar coordinates. Understand the conditions under which f is continuous, focusing on non-negative values of k. Suitable for advanced high school or college students studying calculus.

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