91Ó°ÊÓ

The function is defined as $$f(x, y) = \frac{x y}{x^{2}+y^{2}}$$ when (x, y) ≠ (0, 0). This function is a rational function of x and y, which is continuous as long as the denominator is nonzero. Since x² + y² ≠ 0 for all points (x, y) ≠ (0, 0), f(x, y) is continuous for all points in \(\mathbb{R}^2\) except (0, 0).

Let us check if the function is continuous at (0, 0) by finding the limit as (x, y) approaches (0, 0): $$\lim_{(x, y) \to (0, 0)} \frac{x y}{x^{2} + y^{2}}$$ To compute this limit, let's use polar coordinates, by letting \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\), where \(r \geq 0\) and \(0 \leq \theta \leq 2\pi\). If the limit exists, it should be independent of the value of \(\theta\). Thus, consider the expression as r approaches 0: $$\lim_{r \to 0} \frac{r^2 \cos(\theta) \sin(\theta)}{r^2 (\cos^2(\theta) + \sin^2(\theta))} = \lim_{r \to 0} \cos(\theta) \sin(\theta)$$ For fixed \(\theta\), the limit above will have different values, i.e., the limit depends on \(\theta\). Therefore, the limit $$\lim_{(x, y) \to (0, 0)} \frac{x y}{x^{2} + y^{2}}$$ does not exist. Since the limit does not exist, the function is not continuous at (0, 0).

The function f(x, y) is continuous for all points (x, y) in \(\mathbb{R}^2\) except (0, 0) and is discontinuous at (0, 0).