The theory of functions of a real variable and the theory of Fourier's series: The theory of functions of a real variable and the theory of Fourier's series

Hobson, E. W.

Bibliographic data

Monograph

Persistent identifier:: 191721586X

Title:: Die Provinz Hannover

Sub title:: in Geschichts-, Kultur- und Landschaftsbildern : mit 83 Abbildungen im Text, 5 Vollbildern und einem Doppelbild, sowie einer Karte der Provinz Hannover von C. Diercke

Scope:: XII Seiten, 1686 Spalten, 5 ungezählte Blätter Bildtafel, 2 ungezählte gefaltete Blätter Bildtafeln

Info on language/writing:: In Fraktur

Edition title:: Zweite, vollständig umgearbeitete und wesentlich vermehrte Auflage

DOI:: 10.14463/KXP:191721586X

Year of publication:: 1888

Place of publication:: Hannover

Publisher of the original:: Verlag von Carl Meyer (Gustav Prior)

Identifier (digital):: 191721586X

Illustration:: Illustrationen, Karte

Signature of the source:: G 38-3

Language:: German

Usage licence:: Public Domain Mark 1.0

Editor:: Meyer, Johannes; Diercke, Carl; Ebert, Adolf; Görges, Ernst

Printer:: Meyer, Carl

Publisher:: Carl Meyer, Hannover

Publisher of the digital copy:: Technische Informationsbibliothek (TIB)

Place of publication of the digital copy:: Hannover

Year of publication of the original:: 2025

Document type:: Monograph

Collection:: Geography; Homeland and regional geography

Title:: Inhalt.

Document type:: Monograph

Structure type:: Table of contents

423 324,325] Functions continuous in each variable square x^ ¡3, a ^ y ^ /3, and is continuous at every point with respect to x and with respect to y, and moreover is equal to 0 (x) on the straight line X = yt. (2) What must be the nature of a function (x), defined for a ^ x ^ /3, in order that a function f(x, y) can be defined for all points in the square «<¿£</3, a^y /3, and which shall satisfy the conditions that it is con tinuous with respect to (x, y) at every point for which y > 0, is continuous with respect to y at the points of y = 0, and is equal to cj) (x) when y — 0 ? (3) A function /(x, y) is defined in the rectangle a^x ^ ¡3, y^y^S, and is everywhere continuous with respect to y. Further, there is a set of parallels to the ¿c-axis, along each of which f(x, y) is continuous with respect to x; these parallels intersecting the straight line x = a in a set of points which is everywhere dense in the interval (7, 8). What is the nature of the function f(x, y) on a continuous curve drawn in the rectangle ? The problems (1), (2) are particular cases of (3). It has been shewn above that a necessary condition satisfied by f (x, y), in (3), is that it should be a point-wise discontinuous function relatively to every perfect set of points. That this condition is also sufficient, has been demonstrated by Baire in his memoir quoted above. A proof of this will be given, for the case of problem (2), in Yol. 11, in connection with the theory of functions representable as the limits of sequences of functions. THE REPRESENTATION OF A SQUARE ON A LINEAR INTERVAL. 325. Let a point of a square whose side is unity be denoted by {x, y), where 0^x^l,0^y^l; and let t denote a point of a linear interval (0, 1). An account has been given in § 62 of Cantor’s method of establishing a (1, 1) correspondence between the points of the square and those of the linear interval. Such a correspondence denotes functional relations x = f{t), y = cp(t) between x, y as dependent variables, and t as an independent variable. It will be shewn however that no (1, 1) relation between the two sets of points can be a continuous representation*; i.e. it is impossible that the functions f(t), (p(t) can be both continuous. Let us assume that such a continuous representation can be defined. To any closed set of points {£}, in (0, 1), there will correspond a closed set in the plane area. For if t 1} t 2 , ... t n , ... be a convergent sequence of points t, of which t 0i is the limiting point, then the point /(¿ w ), (fj (A) is the limiting point of the set of points {x-y, y-i), (x 2 , y 2 ),... {oc n ,yb,... which correspond * See Netto, “ Beitrag zur Mannigfaltigkeitslebre,” Crelle’s Jl., vol. lxxxvi. ; also Loria, Giorn. di Mat., vol. xxv, p. 97. In the proof given by these writers it is assumed that a closed curve corre sponds to a linear sub-interval of (0, 1); this is not necessarily the case, for a non-dense closed set may correspond to the closed curve.

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