Question

Find the largest \(\epsilon\) such that \(S\) contains an \(\epsilon\) -neighborhood of \(x_{0}\). (a) \(x_{0}=\frac{3}{4}, S=\left[\frac{1}{2}, 1\right)\) (b) \(x_{0}=\frac{2}{3}, S=\left[\frac{1}{2}, \frac{3}{2}\right]\) (c) \(x_{0}=5, S=(-1, \infty)\) (d) \(x_{0}=1, S=(0,2)\)

Short answer

Question: What is the largest 𝜖 such that 𝑆 contains an 𝜖-neighborhood of 𝑥₀ for the following cases: (a) \(x_{0}=\frac{3}{4}\) and \(S=\left[\frac{1}{2}, 1\right)\) (b) \(x_{0}=\frac{2}{3}\) and \(S=\left[\frac{1}{2}, \frac{3}{2}\right]\) (c) \(x_{0}=5\) and \(S=(-1, \infty)\) (d) \(x_{0}=1\) and \(S=(0,2)\) Answer: (a) The largest 𝜖 is \(\frac{1}{4}\). (b) The largest 𝜖 is \(\frac{1}{6}\). (c) The largest 𝜖 is \(6\). (d) The largest 𝜖 is \(1\).

Step-by-step solution

Identify the interval and point \(x_{0}\).

For this case, we have \(x_{0}=\frac{3}{4}\) and \(S=\left[\frac{1}{2}, 1\right)\).

Calculate the distance to the nearest boundary point.

There are two boundary points in consideration: \(\frac{1}{2}\) and \(1\). Since \(x_{0}\) is closer to \(1\), we find the distance: \(\epsilon = 1-\frac{3}{4}=\frac{1}{4}\).

Largest epsilon value.

The largest \(\epsilon\) is \(\frac{1}{4}\). (b)

Identify the interval and point \(x_{0}\).

For this case, we have \(x_{0}=\frac{2}{3}\) and \(S=\left[\frac{1}{2}, \frac{3}{2}\right]\).

Calculate the distance to the nearest boundary point.

There are two boundary points in consideration: \(\frac{1}{2}\) and \(\frac{3}{2}\). Since \(x_{0}\) is closer to \(\frac{1}{2}\), we find the distance: \(\epsilon = \frac{2}{3}-\frac{1}{2}=\frac{1}{6}\).

Largest epsilon value.

The largest \(\epsilon\) is \(\frac{1}{6}\). (c)

Identify the interval and point \(x_{0}\).

For this case, we have \(x_{0}=5\) and \(S=(-1, \infty)\).

Calculate the distance to the nearest boundary point.

There is only one boundary point in consideration: \(-1\). We find the distance: \(\epsilon = 5-(-1)=6\).

Largest epsilon value.

The largest \(\epsilon\) is \(6\). (d)

Identify the interval and point \(x_{0}\).

For this case, we have \(x_{0}=1\) and \(S=(0,2)\).

Calculate the distance to the nearest boundary point.

There are two boundary points in consideration: \(0\) and \(2\). Since \(x_{0}\) is equidistant from both, we choose either one of them. We find the distance: \(\epsilon = 1-0=1\).

Largest epsilon value.

The largest \(\epsilon\) is \(1\).

Key concepts

Epsilon-Neighborhood

An epsilon-neighborhood is a fundamental concept in real analysis that denotes a set of points within a certain distance from a central point. Specifically, if we take any point, say x₀, the epsilon-neighborhood of x₀ is the set of points that lie within an interval (x₀ - ε, x₀ + ε) where ε is a positive number.

When solving exercises involving epsilon-neighborhoods, understanding this concept helps you to visualize the interval around x₀ which contains all points that are 'close' to x₀. In the given exercise, for instance, the goal was to find the largest epsilon such that the neighborhood is completely contained within the given set S. It's crucial to pick the right epsilon that ensures all points in the neighborhood are inside S without touching or crossing the interval's boundaries.

Boundary Points of an Interval

In the context of interval calculations in real analysis, the term boundary points refers to the 'edges' of intervals. An interval is a contiguous block of numbers within the real number line. The boundary points are those at the very end or very beginning of these blocks.

An interval can be closed or open at its boundaries; a closed interval includes its boundary points, denoted by square brackets [ ], whereas an open interval does not, denoted by parentheses ( ). It's important to identify whether the boundary points belong to the set because it influences the selection of epsilon when seeking an epsilon-neighborhood. For example, in the exercise, the largest allowable epsilon depends on whether the interval is open or closed at the boundary nearest to x₀.

Real Analysis Interval Calculations

Calculations involving intervals are a key aspect of real analysis, often requiring you to determine relations between points and intervals, the length of intervals, or the maximum allowed epsilon for an epsilon-neighborhood.

When performing these calculations, precision is key. You may need to determine whether a point lies within an interval, how far it is from the interval's boundaries, and if those boundaries are included in the set. Use inequalities to express these relationships and check the calculations carefully. In terms of the given exercise, calculating the distance to the nearest boundary point determines the maximum size of the epsilon-neighborhood that can fit within the interval without exceeding it.

Nearest Boundary Distance

The nearest boundary distance is a measure of how far a point is from the nearest end of an interval. It's vital for determining the size of an epsilon-neighborhood in real analysis. The nearest boundary distance is calculated simply by finding the absolute difference between the point in question and each of the interval's boundary points, then selecting the shortest distance.

For example, when working on problems like the ones given, always keep in mind to look at both boundaries, even though an open interval might suggest that the point has 'infinite distance' to that end. You must compare the distances to both ends and choose the smaller one, as this distance will give you the largest possible epsilon for your epsilon-neighborhood, ensuring the entire neighborhood stays within the interval's bounds.