Problem 39 - Calculus Early Transcendentals

Question

At what points of \(\mathbb{R}^{2}\) are the following functions continuous? $$f(x, y)=\left\\{\begin{array}{ll}\frac{x y}{x^{2}+y^{2}} & \text { if }(x, y) \neq(0,0) \\\0 & \text { if }(x, y)=(0,0)\end{array}\right.$$

Step-by-Step Solution

Answer

(For the point (0, 0), we define \(f(0, 0) = 0\).) Answer: The function \(f(x, y)\) is continuous at all points of \(\mathbb{R}^2\).

Step 1: Continuity for \((x, y) \neq (0, 0)\)

For all points \((x, y) \neq (0, 0)\), the function can be written as \(f(x, y) = \frac{xy}{x^2 + y^2}\). Since both the numerator and the denominator are continuous functions at any \((x, y) \neq (0, 0)\), the quotient of these two functions is also continuous. Therefore, for any point \((x, y) \neq (0, 0)\), the function \(f(x, y)\) is continuous.

Step 2: Analyzing continuity at \((0, 0)\)

To check the continuity of the function at \((0, 0)\), we need to find the limit of the function as it approaches \((0, 0)\). In other words, we must determine if: $$\lim_{(x, y) \to (0, 0)} f(x, y) = f(0, 0)=0$$

Step 3: Computing the limit of the function at \((0, 0)\)

We need to determine if the limit exists for \(f(x, y) = \frac{xy}{x^2 + y^2}\) as \((x, y) \to (0, 0)\). Using polar coordinates, we can rewrite \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\), with \(r \geq 0\) and \(0 \leq \theta < 2\pi\). Now, we will rewrite the function in terms of polar coordinates: $$f(r, \theta) = \frac{r^2 \cos{\theta} \sin{\theta}}{r^2 (\cos^2{\theta} + \sin^2{\theta})} = \frac{r \cos{\theta} \sin{\theta}}{\cos^2{\theta} + \sin^2{\theta}}$$ Notice that when we take the limit as \(r \to 0\), the function becomes: $$\lim_{r \to 0} f(r, \theta) = \frac{0}{\cos^2{\theta} + \sin^2{\theta}} = 0$$ The limit is the same for all values of \(\theta\) and equals to zero. Since the limit exists and equals the value of the function at \((0, 0)\), we can conclude that the function is continuous at \((0, 0)\).

Step 4: Conclusion

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

Key Concepts

Limit

When dealing with the concept of a limit, especially in multivariable functions, we focus on how a function behaves as the input values approach a certain point. In the example given, we analyzed the continuity of the function at the origin \((0, 0)\).

For a function \( f(x, y) \) to be continuous at a point,

• The limit of \( f(x, y) \) as \((x, y)\) approaches that point must exist.
• The limit must be equal to the function's value at that point.

By switching to polar coordinates, we simplify the calculation of limits where Cartesian coordinates might be challenging. We express \( x \) and \( y \) in terms of \( r \) and \( \theta \):

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

Then, we examine the behavior as \( r \to 0 \). In this exercise, the limit \(\lim_{{r \to 0}} f(r, \theta)\) was determined to be 0 for all \(\theta\), indicating that the function approaches the same value from every direction. This confirms that the limit exists and matches the function's value at \((0, 0)\). Thus, it's continuous at the origin.

Polar Coordinates

Polar coordinates are a handy mathematical tool for dealing with points in a plane, especially when working with functions involving radial symmetry or circular paths. To convert from Cartesian \((x, y)\) to polar coordinates, we use:

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

Here, \(r\) represents the distance from the origin to the point, and \(\theta\) is the angle formed with the positive x-axis.

In this particular function, using polar coordinates makes it easier to evaluate continuity at \((0, 0)\) because the variability from different directions in Cartesian form is reduced to changes in \(\theta\). By examining the limit as \(r \to 0\), we deduce consistency for all angles \(\theta\), simplifying our analysis. This makes polar coordinates a powerful method for handling limits where path dependency might obscure simple evaluation in Cartesian form.

Continuous Functions

A continuous function is one where small changes in the input result in small changes in the output. For functions of two variables like \(f(x, y)\), continuity means the function doesn't "jump" as \((x, y)\) changes.

In our example, we tested the function's continuity:

• First, at all points except \((0, 0)\): Since the function can be expressed as a quotient of continuous functions, it's continuous everywhere except possibly \((0, 0)\).
• At \((0, 0)\): Using polar coordinates, the function's limit was found to be consistently 0 from every direction. This matches \(f(0, 0)\), confirming continuity.

To check if a function is continuous:

• Ensure the limit exists as you approach the point.
• Verify that this limit equals the function's value at that point.

Here, both conditions are met, ensuring \(f(x, y)\) is continuous everywhere in \(\mathbb{R}^2\). Continuous functions are helpful because they promise predictable behavior across their domains.