Problem 40 - Calculus Early Transcendentals: Pearson New International Edition

Question

Let \(f\), a function of \(n\) variables, be continuous on an open set \(D\), and suppose that \(P_{0}\) is in \(D\) with \(f\left(P_{0}\right)>0\). Prove that there is a \(\delta>0\) such that \(f(P)>0\) in a neighborhood of \(P_{0}\) with radius \(\delta\).

Step-by-Step Solution

Answer

There exists a \(\delta > 0\) such that \(f(P) > 0\) in a neighborhood of \(P_0\).

Step 1: Understand the problem

We are given a continuous function \(f\) at a point \(P_0\) in an open set \(D\), where \(f(P_0) > 0\). We need to show that there exists a neighborhood around \(P_0\) where \(f(P) > 0\).

Step 2: Use the definition of continuity

Since \(f\) is continuous at \(P_0\) and \(f(P_0)>0\), we can use the epsilon-delta definition of continuity. For \(\epsilon = \frac{f(P_0)}{2}\), there exists a \(\delta > 0\) such that for all points \(P\) in \(D\) where \(||P - P_0|| < \delta\), we have \(|f(P) - f(P_0)| < \frac{f(P_0)}{2}\).

Step 3: Analyze the continuity definition

From the previous step, \(|f(P) - f(P_0)| < \frac{f(P_0)}{2}\) implies \(-\frac{f(P_0)}{2} < f(P) - f(P_0) < \frac{f(P_0)}{2}\). This leads to \(f(P_0) - \frac{f(P_0)}{2} < f(P) < f(P_0) + \frac{f(P_0)}{2}\).

Step 4: Simplify the inequality

The inequality from Step 3 simplifies to \(\frac{f(P_0)}{2} < f(P) < \frac{3f(P_0)}{2}\). This means \(f(P) > 0\) for all \(P\) such that \(||P - P_0|| < \delta\).

Step 5: Verify the solution

Given \(f(P_0) > 0\) and the result that \(f(P) > 0\) in a neighborhood of \(P_0\), we have proved the existence of \(\delta > 0\) such that \(f(P) > 0\) in the neighborhood of \(P_0\).

Key Concepts

Epsilon-Delta Definition

The epsilon-delta definition is a formal way to define the concept of continuity for functions. It's a fundamental tool in calculus, serving as a precise way to explain why and how certain functions behave consistently at specific points. To say that a function \(f\) is continuous at a point \(P_0\), we need:

• For every positive number \(\epsilon\), there exists another positive number \(\delta\)
• Such that for all points \(P\) within a distance \(\delta\) from \(P_0\), the difference \(|f(P) - f(P_0)|\) is less than \(\epsilon\)

This means that the value of the function \(f\) at \(P\) should not differ much from its value at \(P_0\), within the bounds of \(\epsilon\). In our specific problem, we use \(\epsilon = \frac{f(P_0)}{2}\) to ensure that \(f(P)\) remains positive nearby \(P_0\). This careful choice of \(\epsilon\) is crucial to prove the existence of a neighborhood where \(f(P) > 0\).

Neighborhood in Open Sets

In mathematical terms, a neighborhood of a point is simply an area surrounding that point. When we talk about functions and continuity, an open set \(D\) plays a key role. It means any point inside the set has some small area around it that also lies entirely within the set.The neighborhood of a point \(P_0\) is defined in terms of the open set and a small radius \(\delta\). In our exercise, we proved that there's a \(\delta > 0\), forming a neighborhood around \(P_0\), ensuring that \(f(P) > 0\) within that area.

• This neighborhood is like a bubble around \(P_0\), small enough that all points \(P\) within this bubble satisfy \(||P - P_0|| < \delta\)
• The function \(f\) remains positive at each of these points

Understanding neighborhoods helps visualize how continuity affects function values around specific points.

Inequalities in Calculus

In the study of calculus, inequalities are frequently used to establish boundaries and conditions involving functions and their behaviors. In our context, we analyze how inequalities reveal properties about a continuous function:When using the epsilon-delta definition, we derive inequalities to understand how \(f\) behaves near \(P_0\). The inequality \[-\frac{f(P_0)}{2} < f(P) - f(P_0) < \frac{f(P_0)}{2}\]can be simplified to tell us about the positivity of \(f(P)\):

• Rearranging gives \(\frac{f(P_0)}{2} < f(P) < \frac{3f(P_0)}{2}\)
• This ensures \(f(P) > 0\), crucial for establishing a positive neighborhood at \(P_0\).

Inequalities provide the framework we rely on to make definite statements about functions, showing how they either stay positive or maintain certain characteristics over specified regions.