At What Points Of R2 Is The Following Function Continuous

The continuity of a function in multivariable calculus is a fundamental concept that determines where a function behaves smoothly without sudden jumps or breaks. When examining the function f(x, y) = (x² + y²)/(x² - y²), it is essential to determine at which points in R² the function is continuous.

To begin, recall that a function of two variables is continuous at a point (a, b) if the limit of the function as (x, y) approaches (a, b) exists and is equal to the value of the function at that point. For rational functions like f(x, y), continuity generally holds wherever the denominator is not zero. That's why, the primary concern is identifying the points where the denominator x² - y² equals zero.

The denominator x² - y² can be factored as (x - y)(x + y). This leads to this product is zero when either factor is zero, which occurs along the lines x = y and x = -y. Thus, the function f(x, y) is undefined and therefore not continuous at every point lying on these two lines.

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So naturally, the function f(x, y) = (x² + y²)/(x² - y²) is continuous at all points (x, y) in R² except those on the lines x = y and x = -y. Simply put, the set of continuity is R² \ { (x, y) : x = y or x = -y }.

It is also useful to consider the behavior of the function near these lines. Still, as (x, y) approaches any point on x = y or x = -y, the denominator approaches zero, which can cause the function to approach infinity or negative infinity, depending on the direction of approach. This behavior confirms the discontinuity at these lines.

To keep it short, the function f(x, y) = (x² + y²)/(x² - y²) is continuous at all points in R² except those lying on the lines x = y and x = -y. This result follows from the fact that the function is a rational function, and rational functions are continuous wherever their denominators are nonzero. The lines x = y and x = -y represent the only points in R² where the denominator vanishes, leading to discontinuities.