Problem 37 - Calculus with Concepts in Calculus

Question

Determine whether \(f\) is continuous on the given region \(R\). \(f(x, y)=\left\\{\begin{array}{ll}e^{-\left(1+x^{2}\right) / y} & \text { for } y \neq 0 \\ 0 & \text { for } y=0\end{array}\right.\) \(R\) is the upper half plane \(y \geq 0\).

Step-by-Step Solution

Answer

Yes, \(f(x, y)\) is continuous on the region \(R\).

Step 1: Understanding Continuity

To determine if a function is continuous at a point, we need to check if the function is defined at that point, if the limit exists as we approach the point, and if the limit equals the function value at that point.

Step 2: Examining the Function on Region R

Given, the region is the upper half-plane where \(y \geq 0\). For \(y e 0\), the function \(f(x, y) = e^{-\frac{1+x^2}{y}}\) is well-defined as it involves exponential functions, which are continuous.

Step 3: Analyzing the Edge of the Region \(y = 0\)

The function is defined as \(f(x, 0) = 0\). We need to check if \(\lim_{y \to 0^+} f(x, y) = 0\). If it holds, then the function is continuous at \(y = 0\).

Step 4: Calculating the Limit as \(y\) Approaches Zero

Compute \(\lim_{y \to 0^+} e^{-\frac{1+x^2}{y}}\). As \(y\) approaches 0 from the positive side, \(\frac{1+x^2}{y}\) approaches infinity. Consequently, since the exponent is negative, \(e^{-\infty} = 0\).

Step 5: Comparing the Limit to the Function Value at \(y = 0\)

The function value at \(y = 0\) is 0 and the limit as \(y\) approaches 0 is also 0. Therefore, \(f(x, y)\) is continuous at \(y = 0\).

Step 6: Conclusion

Since the function \(f(x, y)\) is continuous for \(y e 0\) and the limit matches the function value at \(y = 0\), we conclude that \(f\) is continuous on the region \(R\).

Key Concepts

Upper Half Plane

The concept of the **upper half plane** is fundamental in multivariable calculus. It refers to the set of all points in a coordinate plane where the y-coordinates are greater than or equal to zero. Visually, if you imagine a graph, this is everything above the x-axis and inclusive of the x-axis itself.

• Definition: The upper half plane is denoted as the set of points such that y ≥ 0.
• This area can be thought of as a restriction or domain where a function is evaluated.

In our problem, the function is considered within this domain, making it crucial to analyze limits and values only within these bounds. This ensures we remain consistent with problem conditions and accurately determine continuity.

Limit of a Function

In calculus, the **limit of a function** is essential to understanding behavior near a point of interest. For a function to be continuous at a point, the limit as we approach that point must exist and match the function's value.

• Intuition: The limit helps us predict where the function is heading, regardless of its value at the point.
• Mathematical Expression: For our two-variable function: \[ \lim_{y \to 0^+} e^{-\frac{1 + x^2}{y}} \] As y approaches zero, the fraction approaches infinity, leading to the exponential of a large negative number, hence zero.

Understanding limits help us determine function behavior, especially near "edges" or boundaries like y = 0, ensuring continuity analysis is performed correctly.

Exponential Function

The **exponential function** is one of the most frequently encountered functions in calculus, represented as \(e^{u}\). It is known for its distinctive properties and continuous nature. In the context of our problem:

• Formula: For y ≠ 0, the function is \(e^{-(1 + x^2)/y}\).
• Behavior: It continuously grows or decays based on the sign and magnitude of the exponent.

Key Properties:

• An exponential function with a negative exponent tends toward zero as the exponent increases in magnitude.
• It is fundamentally continuous across its domain, making it easier to study for edges like when y approaches zero.

Knowing these properties clarifies why we expect continuous behavior everywhere except potentially troublesome points.

Function Continuity

**Function continuity** is a crucial criteria determining whether a function smoothly connects without breaks or jumps in its domain. To evaluate if a function \(f(x, y)\) is continuous in its region \(R\), we need to:

• Defined at the Point: Confirm the function is defined everywhere in \(R\).
• Limit Exists: Ensure the limit of the function exists as we approach any point in \(R\).
• Limit Equals Function Value: Check that this limit equals the function's value at that point in \(R\).

In our exercise, comparing the calculated limit when \(y\) approaches zero to the given function value tests for continuity. Since both are zero, the continuity condition is satisfied at this boundary. Hence, the function \(f(x, y)\) remains uninterrupted across its region \(R\), the upper half plane.