Question

Suppose that \(f\) is bounded on an interval \(\left(x_{0}, b\right]\). Using Definition 2.1.10 as a guide, define \(\overline{\lim }_{x \rightarrow x_{0}+} f(x)\) (the right limit superior of \(f\) at \(x_{0}\) ) and \(\underline{\lim _{x \rightarrow x_{0}}+} f(x)\) (the right limit inferior of \(f\) at \(x_{0}\) ). Then prove that they exist. HiNT: Use Theorem 2.1.9.

Short answer

To summarize, given a function \(f\) bounded on the interval \((x_0, b]\), we defined the right limit superior and the right limit inferior as: $$\overline{\lim }_{x \rightarrow x_{0}+} f(x) = \inf_{x_{0} < x \leq b} \sup_{y>x} f(y)$$ $$\underline{\lim _{x \rightarrow x_{0}+}} f(x) = \sup_{x_{0} < x \leq b} \inf_{y>x} f(y)$$ We then proved that both of these limits exist using Theorem 2.1.9. The right limit superior and right limit inferior exist because the function \(f\) is bounded on the interval \((x_0, b]\), and their respective values can be calculated using the infimum and supremum properties.

Step-by-step solution

Define the right limit superior and right limit inferior

Taking inspiration from Definition 2.1.10, let's define the right limit superior and the right limit inferior as follows: $$\overline{\lim }_{x \rightarrow x_{0}+} f(x) = \inf_{x_{0} < x \leq b} \sup_{y>x} f(y)$$ $$\underline{\lim _{x \rightarrow x_{0}+}} f(x) = \sup_{x_{0} < x \leq b} \inf_{y>x} f(y)$$ These definitions imply that we are looking for the inferior (minimum) value of the supremum of the function for all values to the right of \(x_0\) for right limit superior and the superior (maximum) value of the infimum of the function for all values to the right of \(x_0\) for right limit inferior.

Prove the existence of the right limit superior using Theorem 2.1.9

Now we have to prove that both \(\overline{\lim}_{x \rightarrow x_{0}+} f(x)\) and \(\underline{\lim _{x \rightarrow x_{0}+}} f(x)\) exist. For this, we'll use Theorem 2.1.9, which states the existence of the infimum and the supremum for a bounded set. Since \(f\) is bounded on \((x_0, b]\), we know there exists an infimum \(M = \inf_{x_{0} x_1\}\) has a supremum for any such \(x_1\). Then, the set of all such supremums, namely \(\{\sup_{y > x_1} f(y) \mid x_{0} < x_{1} \leq b\}\), is bounded below by \(M\), so it has an infimum. Thus, the right limit superior exists by Theorem 2.1.9.

Prove the existence of the right limit inferior using Theorem 2.1.9

Similarly, consider a value \(x_2\) such that \(x_0 < x_2 \leq b\). Since \(f\) is bounded, the set of values \(\{f(y) \mid y > x_2\}\) has an infimum for any such \(x_2\). Then, the set of all such infimums, namely \(\{\inf_{y > x_2} f(y) \mid x_{0} < x_{2} \leq b\}\), is bounded above by \(M'\), so it has a supremum. Thus, the right limit inferior exists by Theorem 2.1.9. In conclusion, we have defined the right limit superior and right limit inferior of a function at a point \(x_0\) and proved their existence using Theorem 2.1.9. The right limit superior and right limit inferior exist because f is bounded on the interval \((x_0, b]\), and their respective values can be found using the infimum and supremum properties.

Key concepts

right limit inferior

The concept of the right limit inferior focuses on determining the behavior of a function as it approaches a specific point from the right on the x-axis. Essentially, the right limit inferior \(\underline{\lim _{x \rightarrow x_{0}+}} f(x)\) is defined as the smallest value (or infimum) that corresponds to the set of maximum values (or supremum) of the function beyond a particular point, say from \(x_0\).

Here's how it works:

• Start by picking any point \(x\) slightly greater than \(x_0\) within the given interval.
• Find the smallest value that represents the largest possible outputs of the function beyond this point \(x\).
• The right limit inferior is determined by taking the largest of these smallest values as you move closer to \(x_0\).

Hence, it provides insight into the squeezing lower bound of values that the function approaches, when creeping in from the right towards \(x_0\). This concept is invaluable because it offers a consolidated lower value before reaching \(x_0\).

infimum and supremum

Infimum and supremum are foundational concepts in the realm of mathematical analysis, especially when dealing with bounded functions and limit behavior.

In simple terms, here's what they mean:

• The **infimum** of a function refers to the greatest lower bound of its values on a certain interval. You can think of it as the smallest value that is still greater than or equal to the smallest actual function value.
• The **supremum**, on the other hand, is the least upper bound. It describes the smallest value that is still greater or equal to every function output within an interval. The highest "ceiling" if you will.

These concepts help in understanding the extremes that a function can approach without necessarily touching those extreme values directly. It's akin to understanding how low or high a bridge can be without brushing its surface.

bounded function

A function is considered bounded if its values lie within a fixed range on the given interval. This is key when working with concepts like right limit superior and inferior.

When a function \(f(x)\) is said to be bounded on an interval \((x_0, b]\), much like saying:

• There is a lower bound \(M\) such that \(M \leq f(x)\).
• There is an upper bound \(M'\) such that \(f(x) \leq M'\).

In layman's terms, think of a function as being enclosed within a box with a floor and a ceiling.

This property is crucial for assuring the existence of limits because, in a real-world scenario, it indicates that the function won't go off to infinity or suddenly drop without bound, making the prediction and calculation of limit values manageable.

Theorem 2.1.9

Theorem 2.1.9 plays a pivotal role in establishing the existence of limits in bounded functions. According to this theorem:

• Any bounded non-empty set of real numbers has an infimum and a supremum.

This theorem grants certainty about the ability to estimate and define extreme points (infimum and supremum) without ambiguity when a set is bound by some values.

By applying Theorem 2.1.9 to the right limit inferior and superior, we confirm the existence of these limits. As the theorem assures that the relevant sets have defined bounds, it implies the upper and lower limits of the function converge to a particular value.

It simplifies the understanding of how a function behaves as it approaches its boundary points, thereby reinforcing the reliability of the edges that the function can theoretically approach — integral for proving limit superiors and inferiors of bounded functions.