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The function \( f: \mathbb{R}^2 \to \mathbb{R} \) is defined as follows:

\[
f(x, y) =
\begin{cases}
(x^\alpha + y^\alpha) \sin \left( \frac{1}{\sqrt{x^2 + y^2}} \right), & \text{if } (x, y) \neq (0, 0), \\
0, & \text{if } (x, y) = (0, 0).
\end{cases}
\]

This is a piecewise function defined for two cases:
1. When \( (x, y) \neq (0, 0) \), the function depends on \( x^\alpha \), \( y^\alpha \), and a sinusoidal term involving \( \frac{1}{\sqrt{x^2 + y^2}} \).
2. When \( (x, y) = (0, 0) \), the function is defined to be 0.

Would you like to analyze the continuity or differentiability of this function at \( (0, 0) \) or explore its behavior for specific values of \( \alpha \)? Let me know!

Here are 5 related questions:
1. How do you check if the function is continuous at \( (0, 0) \)?
2. What are the possible implications of the sinusoidal term as \( (x, y) \to (0, 0) \)?
3. How would different values of \( \alpha \) affect the function's smoothness?
4. Can this function be differentiable at \( (0, 0) \)? Why or why not?
5. What is the behavior of the function along specific paths, like \( y = 0 \) or \( x = 0 \)?

**Tip:** To check for continuity at a point, evaluate the limit of the function as it approaches the point from various directions.

Let f: R^2 -> R be defined by f(x,y) = {(x^α + y^α) sin (1 / sqrt(x^2 + y^2)), if (x,y) != (0,0); 0, if (x,y) = (0,0)}.

Explore the continuity and differentiability of a piecewise function involving a sinusoidal term defined over R^2. The function's behavior near the origin, (0,0), is analyzed using multivariable calculus techniques. This problem is suitable for undergraduate-level mathematics students.

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