Problem 8 - Analysis 1: Ein Lehr- und Arbeitsbuch für Studienanfänger

Question

Zeigen Sie mit Hilfe des Riemann-Kriteriums, daß die Funktion $$ f(x)=\operatorname{sgn} x=\left\\{\begin{aligned} 1, & x>0 \\ 0, & x=0 \\ -1, & x<0 \end{aligned}\right. $$ integrierbar über \([a, b]\) ist. (sgn ist die Abkürzung für signum \(=\) Vorzeichen).

Step-by-Step Solution

Answer

The function is Riemann integrable over any interval [a, b].

Step 1: Identify the Interval and Function Properties

The function \( f(x) = \operatorname{sgn} x \) is defined for \( x \in \mathbb{R} \) with values 1 for \( x > 0 \), -1 for \( x < 0 \), and 0 for \( x = 0 \). We need to show that \( f(x) \) is Riemann integrable on the interval \([a, b]\).

Step 2: Assess Function's Discontinuities

The function \( f(x) = \operatorname{sgn} x \) has a discontinuity only at \( x = 0 \), where the function jumps from -1 to 1. Check if \( 0 \) lies within the interval \([a, b]\). If not, the function is continuous on \([a, b]\).

Step 3: Apply the Riemann Criterion

The Riemann criterion states that a function is Riemann integrable on an interval if its set of discontinuities has measure zero. In this case, \( f(x) \) only has a single discontinuity at \( x = 0 \). A single point set has measure zero.

Step 4: Conclude Integrability

Since \( f(x) \) satisfies the Riemann criterion condition (having discontinuities of measure zero), it is Riemann integrable over any interval \([a, b]\), whether \( 0 \) is included or not.

Key Concepts

Signum Function

The signum function, often abbreviated as "sgn," is a simple yet fascinating mathematical function. It is defined as follows:

• For any positive number, the signum function produces a value of 1.
• For a negative number, it gives a value of -1.
• At zero, the function outputs 0.

This function helps determine the sign of a given number without giving importance to the actual size of the number. It acts as a kind of switch that tells you whether a number is positive, negative, or zero. This property makes the signum function a fundamental tool in many areas of mathematics, especially in the analysis where the sign of a value is more pertinent than the value itself.

Discontinuity

Discontinuity occurs in functions where there is a sudden jump, break, or value that is undefined within a certain point in the domain. For the signum function, discontinuity presents itself at a notable point:At the point where \( x = 0 \), the signum function \( \operatorname{sgn} x \) skips suddenly from -1 to 1, creating a jump discontinuity. Despite being a single point, this discontinuity is significant because it marks a place of non-smooth transition in the function's graph. In the context of integrability, addressing discontinuities is crucial as they indicate spots where typical calculation becomes complex, yet crucially do not necessarily impact integrability over an interval.

Measure Zero

Measure theory helps us understand sizes of different sets in terms of their 'measure,' which is akin to length, area, or volume. A set is considered to have measure zero if, in a sense, it occupies no space within the interval it's located.For the signum function, the discontinuity at\( x = 0 \)gives rise to a set that contains just one point. Since any single point in real numbers has measure zero, the discontinuity \( \{0\} \) does not contribute a meaningful 'size' or 'length' when considering measure. This insight is essential for applying integration criteria, linking the concept of measure with the ability to integrate despite discontinuities.

Integration Criteria

The Riemann integration criteria offers a method to check a function's integrability over an interval. For a function to be Riemann integrable:

• Its discontinuities must have measure zero.

With regard to the signum function, \( \operatorname{sgn} x \), its sole discontinuity is at \( x = 0 \).This satisfies the integration criteria since a single point \( \{0\} \) has measure zero. Consequently, this characteristic ensures the function is integrable over any interval \( [a, b] \), regardless of whether zero is within that interval. Understanding these criteria helps clarify why certain functions, despite quirks like jumps, remain suitable for integration.