Problem 40 - Calculus Early Transcendentals

Question

At what points of \(\mathbb{R}^{2}\) are the following functions continuous? $$f(x, y)=\left\\{\begin{array}{ll}\frac{y^{4}-2 x^{2}}{y^{4}+x^{2}} & \text { if }(x, y) \neq(0,0) \\\0 & \text { if }(x, y)=(0,0)\end{array}\right.$$

Step-by-Step Solution

Answer

Answer: The given function is continuous for all points in \(\mathbb{R}^{2}\) except \((x,y) = (0,0)\).

Step 1: Analyze the function for points \((x,y) \neq (0,0)\).

Start by examining the function when \((x,y) \neq (0,0)\) which is: $$\frac{y^{4}-2x^2}{y^4+x^2}$$ Since the numerator and denominator are both continuous functions for \((x,y) \neq (0,0)\), the whole function is continuous at these points.

Step 2: Analyze the function at point \((x,y) = (0,0)\).

Now, let's examine the function at \((x,y) = (0,0)\): $$f(0,0) = 0$$ To check continuity at this point, we need to verify if the limit as \((x,y)\) approaches \((0,0)\) is equal to \(f(0,0)\). If the limit exists and is equal to \(f(0,0)\), then the function is continuous at \((0,0)\).

Step 3: Find the limit as \((x,y)\) approaches \((0,0)\).

Let's evaluate the limit: $$\lim_{(x, y) \to (0,0)}\frac{y^{4}-2x^2}{y^4+x^2}$$ To solve for the limit, let's substitute \(x^2 = r\cos\theta\) and \(y^4 = r\sin\theta\) using polar coordinates: $$\lim_{r \to 0}\frac{r\sin\theta-2r\cos\theta}{r\sin\theta+r\cos\theta} = \lim_{r \to 0}\frac{r(\sin\theta-2\cos\theta)}{r(\sin\theta+\cos\theta)}$$ Since \(0 \le \theta \le 2\pi\), we notice that the limit depends on the value of \(\theta\), making the limit non-existent.

Step 4: Determine continuity.

The function is continuous for all points \((x,y) \neq (0,0)\). However, the limit as \((x,y)\) approaches \((0,0)\) does not exist, so the function is not continuous at \((0,0)\). Therefore, the given function is continuous for all points in \(\mathbb{R}^{2}\) except \((x,y) = (0,0)\).

Key Concepts

Functions of Two Variables

Functions of two variables, like the one in the exercise, assign a real number to each point in a plane, often written as \( f(x, y) \). These functions have an important application because they describe surfaces in three dimensions. In our case, the function \( f(x, y) = \frac{y^4 - 2x^2}{y^4 + x^2} \) considers both the variables \( x \) and \( y \) to find the resultant value.

When we analyze such functions, it's crucial to look at each part separately: the numerator describes the difference between some fourth power of \( y \) and twice the square of \( x \), while the denominator ensures we are not dividing by zero (except at points that need special attention, like \((0,0)\)).

Understanding this idea helps us study changes and behavior of complex surfaces, which is directly linked to concepts of continuity and limits.

Polar Coordinates

Polar coordinates are a valuable tool in the context of functions of two variables. Instead of describing a point \((x, y)\) using rectangular coordinates, we use a radius \(r\) and an angle \(\theta\).

For our function, converting to polar coordinates simplifies the approach to analyzing limits as \((x, y)\) approaches \((0,0)\). We substitute using the relationships \(x = r\cos\theta \) and \(y = r\sin\theta\). This transformation can often simplify otherwise complex algebraic expressions as it bundles the variables into one component, \(r\), which encodes the distance from the origin and \(\theta\), which marks the direction.

Using polar coordinates can offer straightforward evaluation paths for limits where regular \(x, y\) coordinates might fail to reveal crucial behaviors.

Limits and Continuity

Concepts of limits and continuity are vital in determining where a function behaves predictably. When a function approaches a particular point, like \((0,0)\) in our case, we seek to understand if it reaches a specific value. This is the essence of limits.

If the limit of \( f(x, y) \) as \( (x, y) \to (0,0) \) equals the value of the function at that point, \( f(0,0) \), then \( f \) is continuous there. Our analysis shows the limit does not exist at \( (0,0) \) because the direction changes the outcome, confirming the function is not continuous at this point.

Continuity ensures a function's predictability across its domain. Knowing that \( f \) is continuous everywhere except \((0,0)\) in \(\mathbb{R}^2\) tells us exactly where special consideration is needed.

Mathematical Analysis

Mathematical analysis provides the foundation for understanding complex behaviors in functions like our \( f(x, y) \). In such cases, analysis involves breaking down each component—like regions where the function is well-behaved and where extra focus is required, such as potential discontinuities.

Through thorough investigation using polar coordinates and continuity tests, mathematical analysis helps identify points of interest and anomalies.

• Examine critical points, ensuring both numerator and denominator's influence is correctly noted.
• Use alternate forms, like polar coordinates, to gain insights on challenging sections.
• Ensure any discrepancies in continuity are visibly understood for clear conclusions.

Mathematical analysis is the bridge that connects our intuition with rigorous verification, making sure our understanding of functions of two variables is both practical and precise.