/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 15 Gegeben sind zwei stetige Funkti... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 15

Gegeben sind zwei stetige Funktionen \(f\), \(g: \mathbb{R} \rightarrow \mathbb{C}\) mit \(f(x)=g(x)\) f眉r alle \(x \in \mathbb{Q} .\) Zeigen Sie \(f(x)=g(x)\) f眉r alle \(x \in \mathbb{R}\)

Short Answer

Expert verified
Question: Prove that if two continuous functions f and g satisfy f(x) = g(x) for all x in the set of rational numbers (鈩�), then they also satisfy f(x) = g(x) for all x in the set of real numbers (鈩�). Answer: By defining a new function h(x) = f(x) - g(x) and showing that h(x) = 0 for all x in 鈩�, we proved that f(x) = g(x) for all x in 鈩�.

Step by step solution

01

Define the function h

We will define a new function h as the difference between the functions f and g: \(h(x) = f(x) - g(x)\). Since both \(f(x)\) and \(g(x)\) are continuous functions, h(x) will also be a continuous function.
02

Show h(x) = 0 for x in Q

We know that \(f(x) = g(x)\) for all \(x \in \mathbb{Q}\), so if we substitute this into the definition of h(x), we find that \(h(x) = f(x) - g(x) = 0\) for all \(x \in \mathbb{Q}\).
03

Use continuity of h(x) to show h(x) = 0 for x in R

Since h(x) is continuous and h(x) = 0 for all \(x \in \mathbb{Q}\), we will use the continuity of h(x) to show that h(x) = 0 for all \(x \in \mathbb{R}\). Let x be an arbitrary real number. We can always find a sequence of rational numbers \({q_n}\) that converges to x, i.e., \(\lim_{n \to \infty} q_n = x\). Since h(x) is continuous, we have \(\lim_{n \to \infty} h(q_n) = h(x)\). However, we know that h(q) = 0 for all q in Q, so \(\lim_{n \to \infty} h(q_n) = 0\). Therefore, \(h(x) = 0\) for all \(x \in \mathbb{R}\).
04

Conclusion

Since we have shown that \(h(x) = f(x) - g(x) = 0\) for all \(x \in \mathbb{R}\), this implies that \(f(x) = g(x)\) for all \(x \in \mathbb{R}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Continuous Functions
Continuous functions are a fascinating concept in mathematics. They essentially provide a way to describe functions that have no breaks, jumps, or holes in their graph. Imagine a function like a smooth curve you can draw without lifting your pen from the paper. That's what we call a continuous function.

A function \( f(x) \) is continuous at a point \( x = c \) if the following three conditions are met:
  • \( f(c) \) is defined.
  • \( \lim_{x \to c} f(x) \) exists.
  • \( \lim_{x \to c} f(x) = f(c) \).
For a function to be continuous everywhere, these conditions must hold for all points in the function's domain. In the context of the given exercise, the continuous nature of the functions \( f \) and \( g \) allows us to conclude that if they are equal on the rational numbers, they must be equal on all real numbers. This is because continuous functions can't jump off track immediately outside a dense subset like the rational numbers.
Real Numbers
Real numbers are the cornerstone of mathematical analysis. They include all the numbers that you would typically think of鈥攕uch as whole numbers, fractions, and irrational numbers like \( \pi \) and \( \sqrt{2} \).

Real numbers can be visualized as points on an endless line. This line encompasses all possible values from negative infinity to positive infinity. Whenever we talk about functions in calculus, we're usually dealing with real numbers, as they provide the most comprehensive framework for continuity.

A critical aspect of real numbers is that they form a "complete" space. This means that every Cauchy sequence of real numbers (a sequence where the numbers keep getting closer together) converges to a real number. This completeness is what allows continuous functions to exhibit their reliable behavior over the real numbers. It's why, in our exercise, a function equal on all rationals extends its equality over all real numbers.
Rational Numbers
Rational numbers are numbers that can be expressed as a fraction where both the numerator and the denominator are integers, and the denominator is not zero. Examples include \( \frac{1}{2} \), \( -3 \), and \( 7 \). They are a subset of real numbers but play an essential role in analysis due to their density in the real number line.

The density of rational numbers means that between any two real numbers, you can always find a rational number. This property is fundamental for many proofs in mathematical analysis, including the exercise we examined.

Though we can't list all real numbers, we can approximate them precisely using rational numbers. In the problem, it is given that functions \( f(x) \) and \( g(x) \) are equal at every rational number. This fact, combined with their continuity, allows us to extend the equality to all real numbers, showcasing the power of rational numbers in analytical continuity arguments.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Betrachten Sie die beiden Funktionen \(f\), \(g: \mathbb{R} \rightarrow \mathbb{R}\) mit $$ f(x)= \begin{cases}4-x^{2}, & x \leq 2 \\ 4 x^{2}-24 x+36, & x>2\end{cases} $$ und $$ g(x)=x+1 $$ Zeigen Sie, dass die Graphen der Funktionen mindestens vier Schnittpunkte haben.

Gegeben ist eine stetige Funktion \(f:[a, b] \rightarrow\) \(\mathbb{R}\). F眉r alle \(x \in(a, b)\) soll es ein \(y \in(x, b]\) geben mit \(f(y)>\) \(f(x)\). F眉r \(a\) soll dies nicht gelten. Zeigen Sie, dass \(f(x) \leq\) \(f(b)\) f眉r alle \(x \in(a, b)\) gilt, sowie \(f(a)=f(b)\). In der englischen Literatur wird diese Aussage auch als \(R i\) sing Sun Lemma bezeichnet. Den Grund f眉r diesen Namen gibt die Abbildung \(9.33\) wieder.

Bestimmen Sie jeweils den gr枚Btm枚glichen Definitionsbereich \(D \subseteq \mathbb{R}\) und das zugeh枚rige Bild der Funktionen \(f: D \rightarrow \mathbb{R}\) mit den folgenden Abbildungsvorschriften: (a) \(f(x)=\frac{x+\frac{1}{x}}{x}\) (b) \(f(x)=\frac{x^{2}+3 x+2}{x^{2}+x-2}\) (c) \(f(x)=\frac{1}{x^{4}-2 x^{2}+1}\) (d) \(f(x)=\sqrt{x^{2}-2 x-1}\)

\bullet Welche der folgenden Teilmengen von \(\mathrm{C}\) sind beschr盲nkt, abgeschlossen und/oder kompakt? (a) \(\quad\\{z \in \mathbb{C}|| z-2 \mid \leq 2\) und \(\operatorname{Re}(z)+\operatorname{Im}(z) \geq 1\\}\) (b) \(\quad\left\\{\left.z \in \mathbb{C}|| z\right|^{2}+1 \geq 2 \operatorname{Im}(z)\right\\}\) (c) \(\\{z \in \mathbb{C} \mid 1>\operatorname{Im}(z) \geq-1\\}\) $$ \begin{aligned} &\cap\\{z \in \mathbb{C} \mid \operatorname{Re}(z)+\operatorname{Im}(z) \leq 0\\} \\ &\cap\\{z \in \mathbb{C} \mid \operatorname{Re}(z)-\operatorname{Im}(z) \geq 0\\} \end{aligned} $$ (d) \(\\{z \in \mathbb{C}|| z+2 \mid \leq 2\\} \cap\\{z \in \mathbb{C}|| z-\mathrm{i} \mid<1\\}\)

\bullet Es soll gezeigt werden, dass ein abgeschlossenes Intervall \([a, b]\) die Heine-Borel-Eigenschaft besitzt. Gegeben ist ein System \(U\) von offenen Mengen mit $$ [a, b] \subseteq \bigcup_{V \in U} V $$ Man sagt, die Elemente \(V\) von \(U\) 眉berdecken \([a, b]\). Betrachten Sie die Menge \(M=\\{x \in[a, b] \mid[a, x]\) wird durch endlich viele \(V\) aus \(U\) 眉berdeckt. \(\\}\). Zeigen Sie: (a) \(M\) besitzt ein Supremum. (b) Das Supremum von \(M\) ist gleich \(b\).
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.