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[Texts]## 26: Real functions |

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.

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.

See also 54C30

- 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.

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

Dr. Vogel's Gallery of Calculus Pathologies.

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:

- Analysis WebNotes by John Lindsay Orr.
- Interactive Real Analysis by Bert G. Wachsmuth.

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

- Here are the AMS and Goettingen resource pages for area 26.

- Using integrals to show that pi isn't 22/7.
- What functions have antiderivatives which are elementary functions ? Citations and long article by Matthew Wiener. (Includes topics in symbolic integration.) Frequently-mentioned integrands with no elementary antiderivative include exp(-x^2), sin(x)/x, x^x, sqrt(1-x^4), and many variants.
- Where to set teeth on an elliptical gear.
- Calculating the antiderivative of sin(x)^ N .
- Using the Intermediate Value Theorem to disallow functions with f^3=identity.
- The Mean Value Theorem, continuity, and differentiability.
- Functions whose derivatives are not continuous.
- Convergence of infinite products
- An interesting calculus problem: which tangent line is closest to the size of the graph?
- Maximizing a sum of sines (of different periods) (really a question of approximating a number by rationals).
- A polynomial in two variables with two local maxima, no minima or saddle points: two mountains without a valley.
- A paint can which can be filled with a finite volume of paint, but which takes an infinite amount of paint to coat its sides!
- Examples of functions just barely integrable.
- Functions with many negative integrals
- Does integrability imply an easy asymptotic bound? (no)
- Application of Green's theorem (Stokes' theorem) to calculating areas and center of mass of a polygon.
- An application of line integrals to computing center of mass, area, etc using Green's (Stokes') theorem.
- Area bounded by a Lissajou curve.
- Reference for asymptotic expansions of integrals.
- Factoring rational functions as composites.
- Formulae for the Lagrange inversion formula (Taylor series of inverse).
- Calculus: careful statment of theorem locating maxima for functions of one variable
- Calculus (multivariable): how to recognize a global optimum?
- Calculus: do similar functions have similar derivatives?
- Calculus: curve yielding equal volumes under two rotations
- Formula for the equation of a curve formed by rotating the graph of a function
- Effect of rotation on the graph of a function
- Average distance between two points in a ball
- Monotonicity of rational functions of several variables.
- What kind of functions satisfy an anti-Lipshitz condition?
- Consequences of strange replacements for Leibniz's formula for differentiation.
- Differentiating the "difference" (f o g^(-1)) of two monotonic polynomials with resultants.
- What corresponds to the Hessian matrix for vector-valued functions?
- How do we define higher-order derivatives of multivariate functions?
- Faà di Bruno's formula for the iterated derivatives of a composite f o g .
- Jensen's inequality, with an application.
- Axioms (and problems) defining fractional derivatives.
- Defining fractional derivatives with (Laplace) transforms.
- Sources for the history of calculus.

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