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# 26: Real functions

## Introduction

Real functions are those studied in calculus classes; the focus here is on their derivatives and integrals, and general inequalities. This category includes familiar functions such as rational functions.

Calculus information goes here, perhaps. It is the express intention to exclude from this site any routine examples or theorems from comparatively elementary subjects such as introductory calculus. However, there are a few gems, some FAQs, and some nice theory even in the first semester course. There are some more subtle topics which don't often make it to a first-year course.

## Applications and related fields

Some elementary calculus topics may likewise be appropriate for inclusion in 28: Measure and Integration, 40: Sequences and Series, Approximations and expansions, and so on. Use of Newton's method is part of Optimization.

Articles which use results from calculus to solve some problem in, say, geometry would be included in that other page.)

Approximation questions may be part of number theory.

Questions about R^n (say) which are more about the underlying space than about functions on it are dealt with in various geometry and topology pages.

## Subfields

• 26A: Functions of one variable
• 26B: Functions of several variables
• 26C: Polynomials, rational functions
• 26D: Inequalities. For maximal function inequalities, see 42B25; for functional inequalities, See 39B72; for probabilistic inequalities, See 60E15
• 26E: Miscellaneous topics. See also 58CXX

Prior to 1959 there was a subject heading "27 - Analysis" in the MSC; in general, papers in that area would now probably be classed in this section (26).

Browse all (old) classifications for this area at the AMS.

## Textbooks, reference works, and tutorials

Calculus textbooks abound; we will not list them here except to mention the mathematician's favorite, Michael Spivak, "Calculus", Publish or Perish, Berkeley CA, 1980, 647pp, ISBN 0-914-09877-2; and the unique "Freshman Calculus" by Robert A. Bonic et al., Heath 1971.

An unusual resource for the calculus teacher: "A century of calculus", edited by Tom M. Apostol et al. Mathematical Association of America, Washington, DC, 1992. Part I: 1894--1968: 462 pp., ISBN 0-88385-205-5; Part II: 1969--1991: 481 pp., ISBN 0-88385-206-3. -- selections from the American Mathematical Monthly.

Textbooks seen often to be primers!

• Boas, Ralph P., Jr.: "A primer of real functions", The Carus Mathematical Monographs, No. 13; Published by The Mathematical Association of America, and distributed by John Wiley and Sons, Inc.; New York 1960 189 pp. MR22#9550
• Smith, Kennan T.: "Primer of modern analysis", Second edition. Undergraduate Texts in Mathematics. Springer-Verlag, New York-Berlin, 1983. 446 pp. ISBN 0-387-90797-1 MR84m:26002
• Krantz, Steven G.; Parks, Harold R.: "A primer of real analytic functions", Basler Lehrbücher [Basel Textbooks], 4; Birkhäuser Verlag, Basel, 1992. 184 pp. ISBN 3-7643-2768-5 MR93j:26013

Szegö, G. P.; Treccami, G.: "What you should know about real-valued functions but were afraid to ask", Nonlinear optimization (Proc. Internat. Summer School, Univ. Bergamo, Bergamo, 1979), pp. 471--486; Birkhäuser, Boston, Mass., 1980. MR82e:58019

A companion for more advanced students is Gelbaum, Bernard R.; Olmsted, John M. H., "Counterexamples in analysis", The Mathesis Series: Holden-Day, Inc., San Francisco-London-Amsterdam 1964 194pp.

Online texts:

The Truth: Bourbaki, N., "Fonctions d'une variable réelle. Théorie élémentaire." Hermann, Paris, 1976. 331 pp. MR58#28327

## Selected topics at this site

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Last modified 1999/05/12 by Dave Rusin. Mail: feedback@math-atlas.org