Problem 23 - Calculus with Concepts in Calculus

Question

Explain why \(f\) is continuous. $$ f(x, y)=x y^{2} $$

Step-by-Step Solution

Answer

The function \(f(x, y) = xy^2\) is continuous because it is a polynomial, which guarantees continuity over all \(\mathbb{R}^2\).

Step 1: Understand the Definition of Continuity

A function of two variables, say \( f(x, y) \), is said to be continuous at a point \((a, b)\) if the following conditions are satisfied: 1. \( f(a, b)\) is defined. 2. \( \lim_{{(x, y) \to (a, b)}} f(x, y) \) exists. 3. \( \lim_{{(x, y) \to (a, b)}} f(x, y) = f(a, b) \). For \(f\) to be continuous everywhere, it must satisfy these conditions at every point \((x, y)\).

Step 2: Check the Function's Definition

The given function is \( f(x, y) = x y^2 \). This is a polynomial in terms of \(x\) and \(y\), which includes terms \(xy^2\). Polynomials in general are continuous functions, meaning that \( f(x, y) = x y^2 \) is continuous for all \((x, y)\) in \(\mathbb{R}^2\).

Step 3: Validate the Limit Exists Everywhere

The limit \( \lim_{{(x, y) \to (a, b)}} f(x, y) = \lim_{{(x, y) \to (a, b)}} x y^2 \) must exist for the function to be continuous. However, as noted before, since \(x y^2\) is a polynomial, this condition is always satisfied, ensuring continuity across the entire plane.

Step 4: Confirm Function Value Equals Limit at Any Point

Finally, we need to ensure \( f(a, b) = \lim_{{(x, y) \to (a, b)}} f(x, y)\). For any point \((a, b)\), since \( x y^2 \) is an elementary polynomial, you can directly substitute \((a, b)\) into \( f(a, b) \), ensuring this equivalence is satisfied everywhere.

Key Concepts

Polynomials

Polynomials are fundamental objects in mathematics that often show up in many branches. A polynomial in two variables, like our function \[ f(x, y) = x y^2, \]is constructed with terms that involve powers of these variables, combined by addition or multiplication. This specific polynomial is relatively simple, as it just involves the product of the variable \(x\) and the square of \(y\).

• Single-variable polynomials, like \( ax^2 + bx + c \), involve powers of a single variable.
• Multi-variable polynomials generalize this idea to more than one variable.
• Both types share properties like being smooth and continuous everywhere in their domain, which is why multivariable polynomials are continuous on \(\mathbb{R}^2\).

By understanding the structure of polynomials, you know that functions like \(f(x, y)\) do not have breaks or jumps, making them inherently continuous. This property plays a crucial role when discussing limits and continuity in function domains.

Limits

In calculus, the concept of limits is key to understanding how functions behave as they approach a particular point. Often represented as \( \lim_{(x, y) \to (a, b)} f(x, y) \), limits help ascertain the value that a function approaches.Limits and continuity are deeply intertwined because for a function to be continuous at a particular point, the limit must exist at that point. This means as both \(x\) and \(y\) get infinitely close to \((a, b)\), the function \(f(x, y)\) approaches a specific value.

• The limit must be the same irrespective of the path chosen to approach the point \((a, b)\).
• If a polynomial function meets these criteria, it is continuous on its entire domain.

This is why for our example, \(f(x, y) = x y^2\), you can always find such limits wherever needed. Since any polynomial inherently satisfies these properties, you confirm the function's continuity effortlessly.

Multivariable Calculus

Multivariable calculus extends the concepts from single-variable calculus, such as derivatives and integrals, to functions with more than one variable.Continuity in multivariable calculus involves checking limits in more dimensions. For instance, with the function \(f(x, y) = xy^2\), we look at how the function behaves not just as \(x\) approaches a point, but as both \(x\) and \(y\) do simultaneously.

• Multivariable functions, like our polynomial \(xy^2\), cover an entire plane instead of a simple line.
• The concept of neighborhoods, or small areas around a point, becomes crucial for understanding continuity.
• In multivariable calculus, continuity implies smoothness in all directions in the plane.

Having this expanded view allows for the deeper analysis of continuous behavior over surfaces and other high-dimensional spaces, aiding in understanding complex physical phenomena and mathematical models.