Problem 38 - Calculus Early Transcendentals

Question

\(29-38\) Determine the set of points at which the function is continuous. $$f(x, y)=\left\\{\begin{array}{ll}{\frac{x y}{x^{2}+x y+y^{2}}} & {\text { if }(x, y) \neq(0,0)} \\ {0} & {\text { if }(x, y)=(0,0)}\end{array}\right.$$

Step-by-Step Solution

Answer

The function is continuous at all points \((x, y) \in \mathbb{R}^2\).

Step 1: Understand the Function Definition

The function is defined piecewise for two cases: \((x, y) eq (0, 0)\) where \(f(x, y) = \frac{xy}{x^2 + xy + y^2}\), and \((x, y) = (0, 0)\) where \(f(x, y) = 0\). Our goal is to determine the set of points at which this function is continuous.

Step 2: Determine Continuity Away from (0,0)

For points \((x, y) eq (0, 0)\), the function is a rational function. Rational functions are continuous everywhere they are defined, except where the denominator is zero. Here, the denominator \(x^2 + xy + y^2\) is zero only at \((x, y) = (0, 0)\). Therefore, the function is continuous at all points except possibly \((0, 0)\).

Step 3: Check Continuity at (0,0)

For continuity at \((0, 0)\), we need \(\lim_{(x, y) \to (0, 0)} f(x, y) = f(0, 0) = 0\). Consider polar coordinates: let \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). The function becomes \(f(r \cos(\theta), r \sin(\theta)) = \frac{r^2 \cos(\theta) \sin(\theta)}{r^2 (\cos^2(\theta) + \cos(\theta) \sin(\theta) + \sin^2(\theta))}\). Simplifying, \(f(r \cos(\theta), r \sin(\theta)) = \frac{r^2 \cos(\theta) \sin(\theta)}{r^2 (1 + \cos(\theta) \sin(\theta))} = \frac{\cos(\theta) \sin(\theta)}{1 + \cos(\theta) \sin(\theta)}\). This limit is independent of \(r\) and equals 0 as \(r \to 0\), satisfying the condition for continuity at \((0, 0)\).

Step 4: Conclude Continuity

Since \(\lim_{(x, y) \to (0, 0)} f(x, y) = f(0, 0) = 0\) and the function is continuous everywhere else except possibly at the origin, the function \(f(x, y)\) is continuous for all \((x, y) \in \mathbb{R}^2\).

Key Concepts

Piecewise Functions

A piecewise function is a function that is defined by different expressions depending on the input. In the case of the given function, it takes on distinct forms for different conditions:

• For points \(x, y\) not equal to \(0, 0\), the function is expressed as \(\frac{xy}{x^2 + xy + y^2}\).
• At the point \(x, y = 0, 0\), the function is defined to be 0.

The concept of piecewise functions is crucial because it allows us to handle complex systems or physical phenomena where a single formula may not be sufficient to describe all scenarios. This can be particularly useful in applications like engineering and physics, where conditions can change dramatically over different regions.

Rational Functions

Rational functions are functions expressed as one polynomial divided by another. In this exercise, the expression \(\frac{xy}{x^2 + xy + y^2}\) represents a rational function everywhere except where the denominator is zero. When dealing with rational functions, continuity is of interest because these functions are continuous everywhere they are defined unless the denominator is zero. This is a major aspect to consider, as ensuring the denominator is not zero allows the function to maintain its continuity. For this particular function, the denominator \(x^2 + xy + y^2\) is zero only at \(x, y = 0, 0\). Knowing the points where the denominator becomes zero helps us determine where the rational function may have discontinuities. Outside these points, rational functions behave smoothly without abrupt changes.

Limits in Polar Coordinates

Using polar coordinates is a helpful strategy when trying to determine the behavior of functions approaching certain points, especially the origin. Polar coordinates allow us to express any point in the plane in terms of \(r\), the distance from the origin, and \(\theta\), the angle relative to the positive x-axis. In this exercise, the conversion to polar coordinates transforms the original complex expression into one dependent on \(r\) and \(\theta\). Here:

• Set \(x = r \cos(\theta)\)
• Set \(y = r \sin(\theta)\)

This leads to simplifying the function form and easily evaluating limits as \(r\) (the radius or distance from the origin) approaches zero. What makes polar coordinates particularly useful is that they help reveal whether the behavior of a function is consistent in all directions as you approach a particular point, like the origin in this case. By confirming the limit is independent of direction and equals the function's value at the origin \(f(0,0) = 0\), we conclude that the function is continuous at that point.