Problem 31 - Calculus Early Transcendentals

Question

\(29-38\) Determine the set of points at which the function is continuous. $$F(x, y)=\arctan (x+\sqrt{y})$$

Step-by-Step Solution

Answer

The function is continuous for points where \(y \geq 0\).

Step 1: Identify the Continuity of Basic Functions

First, consider the basic functions involved in the composition of the given function. The function \(\arctan(z)\) is continuous for all real numbers \(z\). The function \(x\) is a polynomial and is continuous everywhere. \(\sqrt{y}\) is continuous for \(y \geq 0\).

Step 2: Combine Individual Continuities

The function \(x + \sqrt{y}\) involves the addition of \(x\) and \(\sqrt{y}\). Since addition of continuous functions is continuous and both functions are continuous in their respective domains, \(x + \sqrt{y}\) is continuous where both functions are defined, specifically where \(y \geq 0\).

Step 3: Evaluate Continuity of Arctan Composition

Substitute \(x + \sqrt{y}\) into \(\arctan(z)\). Since \(\arctan(z)\) is continuous for all real \(z\) and \(x + \sqrt{y}\) is continuous for \(y \geq 0\), \(F(x, y) = \arctan(x + \sqrt{y})\) is continuous for \(y \geq 0\).

Step 4: Conclusion of Continuity Domain

The function \(F(x, y) = \arctan(x + \sqrt{y})\) is continuous wherever \(x + \sqrt{y}\) is defined and \(\arctan\) can be applied. Therefore, \(F(x, y)\) is continuous at all points where \(y \geq 0\).

Key Concepts

Composition of Functions

Composition of functions involves creating a new function by applying one function to the result of another function. This is expressed as \ \(f(g(x))\ \), meaning you first apply function \ \(g\ \) to \ \(x\ \) and then apply function \ \(f\ \) to the result. In our example, the composition is \ \(\arctan(x + \sqrt{y})\ \).

Here's how it works:

• First, calculate \ \(x + \sqrt{y}\ \) as the interior function. This involves adding the polynomial \ \(x\ \) and the square root function \ \(\sqrt{y}\ \).
• Then, take the arctan of the result, which is the outer function.

This layered application of functions is what forms a composition, and the continuity of the composed function depends on each individual component's continuity. The composed function \ \(F(x, y)\ \) remains continuous where each function in the composition is continuous. Thus, \ \(F\) is continuous for \ \(y \geq 0\ \).

Arctan Function

The arctan function, also known as the inverse tangent function, returns the angle whose tangent is the number you're evaluating. It's denoted by \ \(\arctan(z)\ \).

Key features include:

• Defined for all real numbers \ \(z\ \), meaning it can take any value from the real number set as its input.
• It has a range of \ \([-\frac{\pi}{2}, \frac{\pi}{2}]\ \), providing angles in radians.
• It is known to be continuous everywhere on its domain, which is why it's a core function in many types of composition.

In our function \ \(F(x, y) = \arctan(x + \sqrt{y})\ \), the continuity of \ \(\arctan\ \) ensures that as long as \ \(x + \sqrt{y}\ \) is a real number, \ \(F(x, y)\ \) will be continuous.

Sqrt Function

The square root function, denoted as \ \(\sqrt{y}\ \), is a classic example of a function with a restricted domain. It only accepts non-negative numbers as inputs \ \(\(y \geq 0\)\ \), since the square root of negative numbers is not defined within the real number system.

Characteristics of the sqrt function include:

• It begins at zero, \ \(\sqrt{0} = 0\ \), and extends to infinity as \ \(y\ \) becomes large.
• It's continuous for \ \(y \geq 0\ \), meaning no sudden jumps or breaks in its value as \ \(y\ \) changes.
• Used widely in compositions, like our function \ \(x + \sqrt{y}\ \), its continuity is crucial in determining the overall continuity of composed functions.

So, for continuity considerations in the function \ \(x + \sqrt{y}\ \), we must ensure \ \(y\) is greater than or equal to zero.

Real Analysis

Real analysis is a major branch of mathematics focusing on real numbers and real-valued functions. It explores properties like sequences, series, limits, and continuity—fundamental aspects when analyzing functions.

Key points in real analysis include:

• Understanding how functions behave, particularly around concepts like limits and continuity. This is crucial for determining domains where functions like \ \(F(x, y)\ \) remain continuous.
• Studying how different functions interact through operations like composition, providing insights into whether new functions created are continuous.
• Real analysis is foundational to calculus, which uses these concepts extensively to study changes and areas under curves.

In problems involving functions like \ \(F(x, y) = \arctan(x + \sqrt{y})\ \), real analysis helps us understand under what conditions the outputs are continuous—hence predicting the behavior of these compositions within specified domains.