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 \(\delta\).

Short answer

A neighborhood exists around \( P_0 \) where \( f(P) > 0 \).

Step-by-step solution

Understand the Problem

First, it's important to understand that the function \( f \) is continuous at the point \( P_0 \), which means there are certain properties of continuity we can use. We are tasked with proving there exists a neighborhood around \( P_0 \) such that \( f(P) > 0 \) for all points \( P \) in that neighborhood.

Recall the Definition of Continuity

By the definition of continuity, for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that for all points \( P \) in \( D \): \[ \|P - P_0\| < \delta \Rightarrow |f(P) - f(P_0)| < \epsilon. \] Since \( f(P_0) > 0 \), we want to pick \( \epsilon \) such that \( f(P) \) remains positive.

Choose an Appropriate \( \epsilon \)

Select \( \epsilon = \frac{f(P_0)}{2} \) since it is positive due to \( f(P_0) > 0 \). This choice will help demonstrate that \( f(P) \) is also positive in the neighborhood.

Apply the Continuity Property

From the continuity definition, with \( \epsilon = \frac{f(P_0)}{2} \), there exists a \( \delta > 0 \) such that:\[ \|P - P_0\| < \delta \Rightarrow |f(P) - f(P_0)| < \frac{f(P_0)}{2}. \] This implies \( f(P) \) must be close to \( f(P_0) \).

Show Positivity of \( f(P) \)

The inequality \( |f(P) - f(P_0)| < \frac{f(P_0)}{2} \) leads to:\[ -\frac{f(P_0)}{2} < f(P) - f(P_0) < \frac{f(P_0)}{2}. \]Thus, \( f(P) > f(P_0) - \frac{f(P_0)}{2} = \frac{f(P_0)}{2}. \) Since \( \frac{f(P_0)}{2} > 0 \), it follows that \( f(P) > 0 \) for \( \|P - P_0\| < \delta \).

Conclusion

By the properties of continuity and our choice of \( \epsilon \), we demonstrate that there exists a \( \delta > 0 \) such that for all \( P \) within this distance \( \delta \) from \( P_0 \), \( f(P) > 0 \). This effectively constructs the neighborhood of \( P_0 \) where the function remains positive.

Key concepts

epsilon-delta definition

Understanding the epsilon-delta definition is crucial in grasping the concept of continuity in calculus. This definition provides the backbone of what it means for a function to be continuous at a given point. Here's the essence of the definition: for a function \( f \) to be continuous at a point \( P_0 \), for every \( \epsilon > 0 \), there must exist a \( \delta > 0 \) such that if the distance between any point \( P \) and \( P_0 \) is less than \( \delta \), then the difference between \( f(P) \) and \( f(P_0) \) is less than \( \epsilon \).

• \( \epsilon \) represents how close \( f(P) \) needs to be to \( f(P_0) \).
• \( \delta \) signifies the allowable distance from \( P_0 \) within the domain.

By choosing an \( \epsilon \) value, we dictate how close we want the function's output to stick to \( f(P_0) \). The definition assures us that for this chosen closeness, there's a corresponding distance, \( \delta \), around \( P_0 \) where this condition holds true.

positive neighborhood

A positive neighborhood is a term used to describe a region around a point where the values of a function remain positive. In the context of this exercise, we are interested in finding a neighborhood around the point \( P_0 \) where the function \( f \) is not just continuous but also positive.
This ensures that, given \( f(P_0) > 0 \), there is an area around \( P_0 \) such that \( f(P) > 0 \) for every \( P \) within that area. This concept is demonstrated by choosing an appropriate \( \epsilon \) (like half of \( f(P_0) \) in the provided solution).
By establishing the existence of a \( \delta > 0 \) when \( \epsilon = \frac{f(P_0)}{2} \), we guarantee that:

• \( f(P) \) is close to \( f(P_0) \), remaining above zero.
• This positive neighborhood confirms that the function values continue to stay positive around \( P_0 \).

multivariable functions

Multivariable functions involve scenarios where a function has more than one input. In our exercise, \( f \) is a function of \( n \) variables. Rather than dealing with a single line or curve, multivariable functions can be visualized as surfaces or hypersurfaces in higher dimensions.
They require an understanding of how changes in all of these variables affect the output of the function. This complexity means that the ideas of limits and continuity must be examined in all possible directions and dimensions in the multivariable domain.
Multivariable functions differ from single-variable ones in that:

• The concept of a limit involves considering the approach to a point from multiple directions.
• Continuity needs verification over all these paths in space, ensuring the function doesn't "break" across any of them.

The continuity at \( P_0 \) is critical because it implies that small changes in any direction from \( P_0 \) won't drastically affect the output \( f(P) \).

continuity properties

Continuity properties in calculus elucidate how a function behaves as its input approaches a specific point. For a function to be continuous at a point \( P_0 \), it must meet specific criteria based on the epsilon-delta definition previously discussed.
The role of these continuity properties helps guarantee that:

• The function doesn't have any sudden jumps or interruptions in its value at or near the point.
• Slight changes in input lead to only slight changes in output.
• Specifically for the exercise, they show that there's a reality-based consistency in the function's behavior, allowing for the establishment of a positive neighborhood around \( P_0 \).

These properties are foundational in proving mathematical theorems and ensuring coherent functions in real-world applications like engineering and physics.