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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 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.

History

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.

See also 54C30

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

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

Software and tables

Other web sites with this focus

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

Selected topics at this site

• Statements from Calculus which are more or less equivalent to the completeness of the reals
• Bernoulli inequality proves e exists
• "Natural" example of a function with distinct one-sided limits
• Some less-trivial applications of L'Hospital's Rule
• Proof of L'Hospital's Rule
• The Cantor function
• Convergence of infinite products
• Using the Intermediate Value Theorem to disallow functions with f^3=identity.
• Frechet interpretation of derivatives (linear maps)
• Derivatives have the Darboux property (hence if increasing, are continuous)
• Which functions on R are derivatives?
• A strictly increasing function with a derivative equal to zero on a dense set
• The symmetric derivative and domains of symmetric differentiability
• Functions whose derivatives are not continuous.
• 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.
• The Mean Value Theorem, continuity, and differentiability.
• The role of the Mean Value Theorem
• Importance of the Implicit Function Theorem
• Illustrative examples of everywhere continuous, nowhere differentiable functions
• "Natural" smooth, nowhere analytic functions
• Description and utility of the Hessian matrix
• What is the 2nd derivative test for Lagrange multipliers?
• Functions not constant on a curve of critical points
• 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 .
• 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!
• The can of paint with finite volume but infinite surface area (Gabriel's horn)
• Examples of functions just barely integrable.
• Functions with many negative integrals
• Does integrability imply an easy asymptotic bound? (no)
• Errors in tables of integrals and special functions
• Using the web to check the 'standard' symbolic integrals
• Rewriting trigonometric integrals as algebraic (elliptic) ones
• Tricks for evaluating integrals (add a parameter and differentiate with respect to it)
• Typical evaluation of a definite integral with erf(x) in the integrand
• Why the x=tan(u/2) substitution works for ellipse and integration problems
• Cauchy principal value for integrals (and why to treat it carefully)
• Application of Green's theorem (Stokes' theorem) to calculating areas and center of mass of a polygon.
• Stokes' theorem on surfaces with singularities
• 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 .
• Sequence rapidly converging to the Arithmetic-Geometric Mean of two numbers
• Quasiperiodical functions on R
• Limiting behaviour of subadditive functions on R
• Shapiro's Inequality regarding Sum( a_n/(a_{n-1}+a_{n+1}) )
• Convergence of the derivatives of a convergent sequence of functions
• Convex functions and related notions
• Jensen's inequality, with an application.
• Axioms (and problems) defining fractional derivatives.
• Defining fractional derivatives with (Laplace) transforms.
• Overviews of fractional order derivatives
• Sources for the history of calculus.