Theory Of Real Functions Research Articles

Book Review: Complexity theory of real functions

Book Review: Complexity theory of real functions

Ker-I Ko. Complexity theory of real functions. Progress in theoretical computeer science. Birkhäuser, Boston, Basel, and Berlin, 1991, viii + 309 pp.

Ker-I Ko. Complexity theory of real functions. Progress in theoretical computeer science. Birkhäuser, Boston, Basel, and Berlin, 1991, viii + 309 pp.

An introduction to the theory of real functions

Introduction Functions of Bounded Variation Approximation of Continuous Functions by Polynomials Functions on Metric Spaces Algebras of Sets and Measurable Functions Measure and Measurable Functions Integration Differentiation Appendices Exercises Bibliography Subject Index.

Invariance in calculus

Elementary calculus may be taught as a theory of real functions on the real line. It is then natural to introduce symbols for composition and for the identity function. But there is a general hesitation to do this. The reason is that most applications of calculus actually deal with real functions on a one‐dimensional manifold, and with their differentials. This paper shows how the scalars and differentials of tensor analysis may be presented at this level. The resulting theory is not only simple to understand, but is necessary in order to make any sense out of elementary calculus.

The Elements of the Theory of Real Functions. By J. E. Littlewood Pp. 71. 9s. 6d. 1956. (Heffers, Cambridge. Dover, New York) - Hypothèse du Continu. By W. Sierpinski Pp. 274. $4.95. 1957. (Chelsea, New York)

The Elements of the Theory of Real Functions. By J. E. Littlewood Pp. 71. 9s. 6d. 1956. (Heffers, Cambridge. Dover, New York) - Hypothèse du Continu. By W. Sierpinski Pp. 274. $4.95. 1957. (Chelsea, New York)

The Elements of the Theory of Real Functions. J. E. Littlewood. Third Edition. Dover Publications, New York. 71 pp. $1.35.

The Elements of the Theory of Real Functions. J. E. Littlewood. Third Edition. Dover Publications, New York. 71 pp. $1.35.

Henri Lebesgue, 1875 - 1941

Henri Léon Lebesgue was born in 1875 and died in 1941. He studied at the Ecole Normale Superieure, and his first post appears to have been that of maitre des conferences at Rennes, which he held until 1906. He then moved to Poitiers, where he was described first as ‘charge de cours a la faculte des Sciences’ and later as professor. In or about 1912 he was called to Paris as maitre des conferences and he afterwards became professor at the College de France. He was elected to the Academie des Sciences in 1922. He was made an honorary member of the London Mathematical Society in 1924, and a foreign member of the Royal Society in 1934. Towards the close of the last century, the development of mathematical analysis may be said broadly to have reached the stage at which a piece of work wherein only continuous functions were encountered could be carried through. For example, the Riemann integral solved the problem of finding a function having a given derivative if the derivative was continuous. Again, everything was known about the length of a curve and its expression as an integral if it had a continuously turning tangent. Artificial and unaesthetic restrictions had repeatedly to be made to keep out discontinuous functions. Jordan, in the preface to his Cours d'Analyse (2nd edition, 1893), wrote: ‘Certains points presentent encore quelque obscurite. Pour en citer un exemple, nous n’avons pu arriver a definir d’une maniere satisfaisante l’aire d’une surface gauche, que dans le cas ou la surface a un plan tangent, variant suivant une loi continue’. In 1898 vital steps were taken by Baire and by Borel. The thése of Baire was a systematic and penetrating discussion of discontinuous functions. And Borel in the small compass of four pages of his tract, Leçons sur la théorie des fonctions , propounded his theory of measure, fundamentally more powerful than Jordan’s theory of content in being additive for an enumerable infinity of sets. All was ready for the rapid advances of the next decade which made the theory of real functions into a satisfying whole; in this transformation the leading part was played by Lebesgue.

Methods in point sets and the theory of real functions

Methods in point sets and the theory of real functions

The Elements of the Theory of Real Functions

The Elements of the Theory of Real Functions

On the foundations of the calcul fonctionnel of Fréchet

In his thesis, t Frechet gave a very beautiful generalization of the theory of point sets and of the theory of real valued functions of a real variable. His functions are real valued but the range of the independent variable is an abstract class -S of elements q . He secures his results principally through the medium of a properly conditioned distance function 8, a generalization of the distance between two points, which associates with each pair q1 q2 of elements a real non-negative number 8 (qi q2 ) . He is thus enabled to secure the more important theorems of point set theory and of real function theory, especially those relating to continuous functions and their properties. This theory of Frechet has excited considerable interest and has received much attention from mathematicians. Various contributions to its foundations and to its content have been made. In the present paper we follow the example of Frechet in assuming once for all that 8 (qq) = 0 and that 8 (qi q2 ) = 8 ( q2 q ) . In other words we assume that the distance from an element to itself is zero and that the distance between two elements is independent of the order in which they are taken. In the first part of the paper we give very simple conditions on systems ( ; 8) which are sufficient for many purposes and which, in the case of compact sets, we show to be equivalent, so far as limit of a sequence is concerned, to the voisinage and thus to the e'cart of Frechet.t The remainder of the paper is devoted to the theory of functions on the sets Z of systems (Z; 8). In terms of the conditions already mentioned we generalize the results of Frechet and Hahn as to the existence of non-constant continuous functions. We give two very