Question

At what points of \(\mathbb{R}^{2}\) are the following functions continuous? $$S(x, y)=\frac{2 x y}{x^{2}-y^{2}}$$

Short answer

Answer: The function S(x, y) is continuous on \(\mathbb{R}^{2}\) excluding the points lying on the lines y = x and y = -x.

Step-by-step solution

Find the Domain of the Function

We must first determine the domain of the function by finding the values of x and y for which S(x, y) exists. The function will be undefined when the denominator is equal to zero: $$x^{2} - y^{2} = 0$$ The equation can be factored as follows: $$(x+y)(x-y) = 0$$ There are two cases when the denominator is zero: 1. x + y = 0 or x = -y 2. x - y = 0 or x = y These are the values of x and y where the function S(x, y) is not defined (undefined).

Continuity at Other Points

Now, we must examine the continuity of the function at points where S(x, y) is defined. Since we have a rational function (a ratio of polynomials), it is continuous everywhere on its domain. Therefore, S(x, y) is continuous at every point (x, y) which satisfies: 1. x ≠ y 2. x ≠ -y

List the Continuous Points

Now that we know the continuity conditions for the function, we can list the points of \(\mathbb{R}^{2}\) where the function S(x, y) is continuous: 1. x ≠ y, or the set of all points (x, y) not lying on the line y = x. 2. x ≠ -y, or the set of all points (x, y) not lying on the line y = -x. Thus, the function S(x, y) is continuous on \(\mathbb{R}^{2}\) excluding the points lying on the lines y = x and y = -x.

Key concepts

Domain of a Function

Understanding the domain of a function is fundamental in calculus, especially when analyzing continuity. The domain of a function consists of all the input values (often 'x' values) for which the function is defined. In simpler terms, it's the set of all possible x-values that won't cause any issues, like division by zero or taking the square root of a negative number in the context of real numbers.

For rational functions, the domain is all real numbers except those that make the denominator zero. Identifying the domain helps avoid undefined behavior in functions and is the first step toward understanding where a function can be continuous. For example, in the function \(S(x, y)=\frac{2 x y}{x^{2}-y^{2}}\), the domain is restricted due to the denominator. The equation \(x^{2} - y^{2} = 0\) implies that we exclude values where \(x = \pm y\), because these would make the denominator zero, thus undefined. Appreciating the domain allows students to better grasp the concept of continuity linked directly to the function's defined region.

Rational Functions

Rational functions are ratios of polynomials, like \(S(x, y)\) in the example, where we have a polynomial in the numerator and another in the denominator. A major characteristic of these functions is how they behave around values that are not included in their domain - this can lead to vertical asymptotes, holes, or undefined points in their graph.

Rational functions are typically discontinuous at points where the denominator equals zero. However, excluding these points from consideration, they are continuous everywhere else on their domain. This puts an emphasis on the importance of understanding not just the concept of functions but also of how they behave over different intervals and the significance of their constraints.

Limits and Continuity

The terms 'limits' and 'continuity' are intertwined in calculus. A function is continuous at a point if the limit of the function as it approaches that point exists and is equal to the function's value at that point. It's about predicting function values as you get infinitely close to a certain x-value without any unexpected jumps or gaps.

A pivotal piece in calculus puzzles, continuity requires a function to be defined at a point, the limit as it approaches the point must exist, and the two values should be the same. When evaluating rational functions like \(S(x, y)\), if they are defined and smooth (no abrupt changes in value) at a particular point, then they are continuous there. However, the catch is that limits and continuity can only be discussed for points within the domain. For \(S(x, y)\), this means everywhere but where \(x = y\) or \(x = - y\), as these are the breach points on the fabric of S’s domain where continuity is disrupted.