Problem 92 - Calculus 3

Question

Discuss the continuity of the following functions. Find the largest region in the \(x y\) -plane in which the following functions are continuous. $$ f(x, y)=e^{3 x y} $$

Step-by-Step Solution

Answer

The function is continuous everywhere in the entire \( xy \)-plane \( \mathbb{R}^2 \).

Step 1: Understand Exponential Functions

The function given is an exponential function, specifically, it involves the expression \( e^{3xy} \). The exponential function \( e^z \) is continuous everywhere for any real number \( z \). This means wherever the argument \( 3xy \) is defined and real, the function is continuous.

Step 2: Define the Domain of the Variables

The expression \( 3xy \) involves the product of \( x \) and \( y \). Since both \( x \) and \( y \) are independent real numbers, their product \( 3xy \) is also a valid real number over all \( x, y \in \mathbb{R} \). Therefore, the expression \( 3xy \) is defined for all points \((x, y)\) in the \( xy \)-plane.

Step 3: Determine the Largest Region of Continuity

Since an exponential expression is continuous wherever its argument is defined (in this case \( 3xy \)), and since \( 3xy \) is defined for any real numbers \( x \) and \( y \), the function \( f(x, y) = e^{3xy} \) is continuous over the entire \( xy \)-plane, \( \mathbb{R}^2 \).

Key Concepts

Exponential Functions

Exponential functions are a core concept in mathematics that are known for their unique and powerful properties. At its simplest, an exponential function is one where the variable is in the exponent, typically in the form \( e^z \), where \( e \) is Euler's number, approximately 2.718. \( e^z \) is distinguished by its continuous growth and its role in various natural phenomena, including compound interest, population growth, and radioactive decay.
In terms of mathematical properties, exponential functions are continuous everywhere. This means there are no breaks, jumps, or holes in the graph of the function for any real number value of the argument \( z \). This continuity property is one of the reasons exponential functions are so vital in both theoretical and applied mathematics.
Furthermore, since the exponential function maintains the same characteristics regardless of the complexity of its argument, it can accommodate complex expressions like \( e^{3xy} \). This tells us that as long as the input \( 3xy \) is well-defined, the function will remain continuous.

Real Numbers

The set of real numbers, denoted \( \mathbb{R} \), encompasses all the numbers along the continuous number line. This includes all the rational numbers (such as integers and fractions) and irrational numbers (such as \( \sqrt{2} \) and \( \pi \)).
Real numbers are essential in defining techniques and solutions across various mathematical domains. They form the basis for analyzing functions, especially those involving continuous processes like calculus.

• Real numbers are crucial because they provide a complete and ordered field, meaning any number between two real numbers is also a real number, and they can be arranged or sequenced.
• In the context of functions, especially exponential functions like \( e^{3xy} \), the real numbers allow for the smooth definition and evaluation over entire planes like the \( xy \)-plane.

Having both \( x \) and \( y \) as real numbers means that the value of \( 3xy \) will also be a real number. This ensures the exponential function remains valid and continuous over the entire region where \( x \) and \( y \) are defined.

Domain of Variables

A function's domain refers to the set of input values for which the function is defined. In many cases, understanding the domain is crucial for analyzing the continuity and behavior of the function.
For the exponential function \( f(x, y) = e^{3xy} \), the variables \( x \) and \( y \) are independent and real, which leads to their product \( 3xy \) being a real number for any real \( x \) and \( y \). This product does not impose any additional restrictions:

• Since real numbers include all positives, negatives, and zero, \( x \) and \( y \) can take any value along the real number line without affecting the continuity of the function.
• Thus, the function \( f(x, y) \) is defined for all pairs \((x, y)\) in the \( xy \)-plane, making \( \mathbb{R}^2 \) its domain of continuity.

The concept of a domain is necessary when determining where a function like this is continuous. Understanding that for \( e^{3xy} \), the domain encompasses all pairs of real numbers provides assurance of its continuity across the entire plane.