Problem 90 - 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. $$ f(x, y)=\sin (x y) $$

Step-by-Step Solution

Answer

The function \( f(x, y) = \sin(xy) \) is continuous over the entire \( xy \)-plane, \( \mathbb{R}^2 \).

Step 1: Understand the function

The function given is \( f(x, y) = \sin(xy) \). Sine functions are known to be continuous everywhere over real numbers. Thus, we need to check this continuity in terms of the variables \( x \) and \( y \).

Step 2: Identify the domain of the function

The sine function \( \sin(xy) \) takes as input the product \( xy \), which can be any real number. There are no restrictions on \( x \) or \( y \) since the product \( xy \) will also be a real number for any \( x, y \in \mathbb{R} \).

Step 3: Determine the continuity

Since \( \sin(z) \) is continuous for all \( z \in \mathbb{R} \) and \( z = xy \) is just a real number, \( f(x, y) = \sin(xy) \) will be continuous on the entire \( xy \)-plane, taking the entire real numbers as input.

Step 4: Define the largest region of continuity

The largest region in the \( xy \)-plane where \( f(x, y) = \sin(xy) \) is continuous is all of \( \mathbb{R}^2 \). Since there are no restrictions or discontinuities, the function is continuous everywhere on the plane.

Key Concepts

Sine Function

The sine function, represented by \( \sin(x) \), is a periodic function that is fundamental in trigonometry. It describes the ratio of the length of the side opposite to an angle in a right triangle over the hypotenuse. The sine function has a smooth, wave-like pattern, oscillating between -1 and 1 over the "x" values. It repeats every \(2\pi\), which is known as its period.
The continuity of the sine function lies in its well-behaved nature. Across the entire range of real numbers \(\mathbb{R}\), the sine function smoothly transitions through its values, never experiencing jumps, gaps, or disruptions. This aspect makes it continuous, allowing it to smoothly describe periodic phenomena in science and engineering without any abrupt changes.

Real Numbers

Real numbers are the foundation of almost all mathematical calculations. They include all the numbers that can exist on the number line, encompassing both rational numbers (like integers and fractions) and irrational numbers (such as \(\pi\) and \(\sqrt{2}\)).
In the context of functions, real numbers allow inputs and outputs that cover a vast array of values. This universality makes them crucial for describing continuous functions like \(\sin(xy)\).

• Real numbers are denoted as \(\mathbb{R}\).
• They include both positive and negative integers, decimals, and fractions.
• They form an ordered set, allowing us to determine which of any two numbers is larger.

Using real numbers ensures that functions such as \(\sin(xy)\) remain defined and continuous across all possible values of \(x\) and \(y\). This seamless nature makes them indispensable for calculus and real-world measurements.

Domain of a Function

The domain of a function is the set of "input" values for which the function is defined. For the sine function \(\sin(xy)\), determining its domain involves identifying all possible pairs of \(x\) and \(y\) that make \(xy\) meaningful as an argument for sine.
Since \(xy\) results in a product that is just another real number, the domain of \(\sin(xy)\) includes all pairs \((x, y)\) such that both \(x\) and \(y\) are real numbers. This results in the entire real number plane \(\mathbb{R}^2\) being the domain.

• The sine function itself does not impose any restrictions on its input.
• A function with no domain restrictions is continuous across its entire domain.
• The importance of identifying a function's domain is ensuring we recognize where the function is valid and operational.

Understanding the domain is crucial in calculus and algebra, helping to map out where a function can be applied and ensuring accurate real-world modeling.