Document Zbl 0070.05203

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New York: McGraw-Hill Book Company, Inc. xii, 375 p. (1956). </div> <div class="abstract">F眉r Kenner der ersten Auflage (New York 1946): Die vorliegende zweite Auflage unterscheidet sich von der ersten nur durch die Anf眉gung der Kapitel XIII und XIV.<br class="zbmathjax-paragraph">Kapitel眉berschriften: I. Einf眉hrung (16 Seiten); II. Das reelle Zahlensystem (23); III. Punktmengen (14); IV. Funktionen und ihre Limiten, Eigenschaften stetiger Funktionen (15); V. Grundlegende S盲tze der Differentialrechnung (16); VI. Das Riemannsche Integral (13); VII. Gleichm盲脽ige Konvergenz (36); VIII. Implizite Funktionen (19); IX. Gew枚hnliche Differentialgleichungen (22); X. und XI. Das Lebesguesche Integral (86); XII. Das Stieltjessche Integral (37). XIII. Mengenlehre und transfinite Zahlen (38); XIV. Metrische R盲ume (36); Sachverzeichnis (5).<br class="zbmathjax-paragraph">In I werden die Grundbegriffe und die im Buch zwar h盲ufig, aber nicht systematisch verwendete logistische Zeichensprache eingef眉hrt, II stellt das System der reellen Zahlen als einen vollst盲ndigen linear geordneten algebraischen K枚rper heraus. In V wird u.a. das Differential von $f$ als Funktion von 2 Argumenten eingef眉hrt, $df(x;\xi)$, welches in $\xi$ linear ist. VII bringt das wichtigste 眉ber unendliche Reihen, Vertauschungss盲tze und unstetige Funktionen. In VIII dienen Fixpunkts盲tze, insbesondere der Brouwersche, zum Existenznachweis f眉r L枚sungen. IX behandelt Systeme von gew枚hnlichen Differentialgleichungen und diesbez眉gliche S盲tze 眉ber die Differenzierbarkeit der L枚sungen und Existenz von ,,ersten Integralen”. In X und XI wird das Lebesguesche Integral nach dem Vorgang von F. Riesz entwickelt: Die me脽baren Funktionen sind die nahezu gleichm盲脽igen Limiten von Folgen von Intervalltreppenfunktionen, das Lebesguesche Integral ist der Limes der Integrale von solchen Folgen. XII enth盲lt eine sehr ausf眉hrliche Analyse des Stieltjesschen Integrals und als Abschlu脽 den Rieszschen Satz 眉ber stetige lineare Funktionale. In XIII definiert Verf. im Anschlu脽 an von Neumann die Mengen $0 = o$, $1 = \{o\}$, $2 = \{o, \{o\}\}, \ldots$ als Ordinalzahlen, welche allgemein auf rein axiomatische Weise erkl盲rt werden, als Mengenmengen $\alpha$, welche (1) verm枚ge der $\in$ Relation wohlgeordnet sind und (2.) die Bedingung ,,Aus $x\in \alpha$ folgt stets $x\subset \alpha$” erf眉llen, und leitet daraus die wichtigsten S盲tze 眉ber Ordinal- und Kardinalzahlen ab. Die verschiedenen Formen und die Rolle des Auswahlaxioms werden besprochen. XIV bringt eine Einf眉hrung in die Topologie der metrischen R盲ume.<br class="zbmathjax-paragraph">Im Buch finden sich zahlreiche 脺bungsaufgaben, unter welchen dem Ref. eine besonders bemerkenswert schien: Sie verlangt, aus 26 ,,Pr盲missen” und 80 ,,Folgerungen” durch Kombination richtige S盲tze bzw. Definitionen zu bilden. Das ganze stellt ein Buch vor, aus dem man in gediegener Weise nicht nur die Theorie der reellen Funktionen sondern auch ein gutes St眉ck modernen mathematischen Denkens lernen kann, und dies an konkreten klassischen Gegenst盲nden.<div class="reviewer"> Reviewer: <a href="/authors/?q=G%2A+Aumann">G. Aumann</a></div> <div class="clearfix"></div><div class="scan"> <span> Page: <a class="goto" href="/scans/070/047.gif">−5</a> <a class="goto" href="/scans/070/048.gif">−4</a> <a class="goto" href="/scans/070/049.gif">−3</a> <a class="goto" href="/scans/070/050.gif">−2</a> <a class="goto" href="/scans/070/051.gif">−1</a> <a class="goto" href="/scans/070/052.gif">±0</a> <a class="goto" href="/scans/070/053.gif">+1</a> <a class="goto" href="/scans/070/054.gif">+2</a> <a class="goto" href="/scans/070/055.gif">+3</a> <a class="goto" href="/scans/070/056.gif">+4</a> <a class="goto" href="/scans/070/057.gif">+5</a></span> <a class="scan" href="/scans/070/052.gif">Show Scanned Page</a><br> <img src="/scans/070/052.gif" alt="scan of Zbl 0070.05203" onclick="$(this).siblings('.scan').click();"> </div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A3116744">Cited in <strong>1</strong> Review</a></div><div class="clear"><a href="/?q=rf%3A3116744">Cited in <strong>57</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A26-01" title="MSC2020">26-01</a> </td> <td class="space"> Introductory exposition (textbooks, tutorial papers, etc.) pertaining to real functions </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A34-01" title="MSC2020">34-01</a> </td> <td class="space"> Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Adifferentiation+and+integration%2C+ordinary+differential+equations">differentiation and integration, ordinary differential equations</a>; <a href="/?q=ut%3ARiemann+integral">Riemann integral</a>; <a href="/?q=ut%3AStieltjes+integral">Stieltjes integral</a>; <a href="/?q=ut%3Auniform+convergence">uniform convergence</a>; <a href="/?q=ut%3Aimplicit+functions">implicit functions</a>; <a href="/?q=ut%3Asets+and+transfinite+numbers">sets and transfinite numbers</a>; <a href="/?q=ut%3Atopology+of+metric+spaces">topology of metric spaces</a></div><div class="keywords"> <h3>Citations:</h3><a href="/0063.01720">Zbl 0063.01720</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 0070.05203" data-ciurl="" data-biburl="/bibtex/03116744.bib" data-amsurl="/amsrefs/03116744.bib" data-xmlurl="/xml/03116744.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/03116744.pdf" title="Zbl 0070.05203 as PDF">Review PDF</a> </div> <div class="fulltexts"> </div> <div class="sfx" style="float: right;"> </div> </div> </article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; 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