Introduction. The Riemann surfaces are a compromise between the continuity and the multiformity. At the Anglo-Saxon Mathematicians, The uniformity or the univocity defines at the same moment a function and its injectivity. At the French-speaking Mathematicians, a function, better an application is injective, surjective and bijective. In the Anglo-Saxon mathematical works the terms of injection of surjection and of bijection are non-existent. The used term is  » one-to-one « . At the Slavic or Russian Mathematicians, such a definition of a function is redundant. In their textbooks, they omit the definition of a surjection. If intuitively, the surjection reminds a topological border, the injection, its inside, to fill the gap, it seemed imperative to define a mathematical object corresponding to a topological outside (ku in kikongo language). As a result, the definition of a function based on the relational univocity of variables accused gaps. To surmount them, it was necessary to violate the rule of the uniformity and to open the way to the said functions multiforms, considered as collections of uniform functions. Every element of this collection becomes a branch of the multivoque function. This hard labour, stammering of the set theory, ended in the Riemann Surfaces.

1. Function. A uniform function f is by definition injective. To two different variables of departure x and y correspond to it two different variables of arrival f(x) and f(y). The Anglo-Saxon Mathematicians omit the definition of the prédicament of the injection attached to the function by the French-speaking Mathematicians. The uniformity or the univocity defines at the same moment a function and its injectivity to the Anglo-Saxon. These last ones use the expression  » One-to-one « , translated into French, univoque or unambiguous. A function f is one-to-one if :
a) f(x) = f(y) a x = y ;
b) (x≠ y) a f(x) f(y).

Now such a formulation of the univocity returns to the definition of an injective function to the French-speaking Mathematicians.

2. Application. Historically, in a care of exactness the notion of function, vague, was replaced by the notion of application.
« A function of the variable x is not inevitably defined for all the values of the variable x » (Jacqueline Lelong-Ferrand, Les notions de mathématiques de base dans l’enseignement du second degré, Paris, Librairie Armand Colin, 1964, p. 33).

The introduction of a domain of definition more the whole departure and an image more the whole arrival clarified the movements of the variables of a function. An application, as the sending of a letter by post, implies a place of expedition and a place of destination. We cross from function to the application by plunging the functions into both sets already quoted. Let be the usual notation of the function :

f : x |→ y.

Domain of f or dom(f) := { x/$y and (x,y) f }.
Range of f or ran(f) := { y/$x and (x,y)  f }. 
The axiom of replacement of the axiomatic set theory according to Zermelo-Fraenkel grants the quality of a set to the range of a set X by an application f :
f(X):= {f(x) / x ∈ X}.

To represent an application f of A in B, we write  f ⊂ A x B (read :  » f is included in A cross B « ); In fact, it is a function f with values in B, and the domain is A. A: = Dom(f) and  ran(f) ⊂ B.  Into is the Anglo-Saxon translation of the preposition dans (mu in kikongo language). An application f of A on B is a function f with values on B, and among which A: = Dom(f) and ran(f):= B. Onto is the Anglo-Saxon translation of the preposition sur (ga in kikongo language).

3. Surjection. Whatever y an element of arrival of a set B, there is at least an element of departure x belonging in a set of departure A. We so define an application or a function surjective. But such a definition is redundant for the Slavic or Russian Mathematicians. In their textbooks, they omit the definition of a surjection. By reminding however the uniqueness of it or of f(x), it returns to a simple définition of a function :

 » We call function […] the binary relation f if for all x, y and z there is of (x, y) ∈ f and (x, z) ∈ f that y = z  » (L. Koulikov, Algèbre et Théorie des nombres, Moscou, Editions Mir, 1982, p. 51).

Otherwise told in every variable of departure corresponds a unique variable of arrival. The definition of a function implies that of its uniqueness. A function is surjective if in any variable of arrival corresponds at least a variable of departure.

4. Multiform functions. Surjection can hide multiform functions in which a variable of departure corresponds to several images of arrival. Notably, the nth roots functions, the logarithmic functions associate with any complex number several roots or several images. They are multiforms there.They are not continuous in all the set of complex numbers. How to work at the same moment on their continuity and on their uniformity ? « On grounds of state », it is necessary to sacrifice the continuity to work on the uniformity. We cannot speak about uniqueness of a collection of arrival, but about several collections of arrival because the idea of continuity of a function is questioned in that case of multiformity. To look after this handicap, we are going to consider a multiform function as a set (in the cantorien sense) of uniform functions. There are so many uniform functions as variables of arrival. Every surjection being then considered as a branch of the multiforms function. The first uniform function will be called main determination. If by leaving a point of origin placed on the main determination, we make several trigonometric tours, in being reviewed in every tour the values of every branch of the multiform function and what we fall again on the initial values of the first main function, this point of origin is called point of connection. Geometrically it is of use as reunification to the various uniform functions, considered as discontinuous leaves. But we are not at the end of the continuity. It is necessary to forbid at first the crossing from a branch to the other one, the change of connection all in all, by introducing a cut on the first main determination. Let us suppose two different points a and b, the one situated at the edge of the line of opening and the other one placed more inside this line taken away from the first one. After a notch, a divides into a’ and a » and b into b’ and b ». We obtain four arranged different points so that a’ is separated of a » and b’ separated of b ». Thanks to the cut, we so win in uniformity of a single branch of the multiform function. We realize a notch on the second branch which gives onto an edge two aligned points c’ and d’ separated respectively from two other aligned points c » of d ». This cut will be realized in every respective branch so returning the separate leaves. The German Bernhard Riemann (1826-1866), student of Carl Gauss (1777-1855) in Göttingen, is going to imagine the connecting of an edge a’ of the opening of the first leaf in the edge c » set by the opening of the neighboring leaf, as well as the connecting of the edge b’ at the edge d ». Let us say edges a » and b » are linked with edges c’ and d’. We can so cross all the leaves in a continuous way, of a’ in c » then of a » in b’; of b’ in o, the center; then of the center in b » towards d’; then from d’ to c’, passing of branch in branch of the function without going round in a circle on a single leaf. The rectangular bands a »b » o d’ c’ and a’ b’ o d » c » form an intertwining. It is all these leaves that constitutes a Riemann surface.

Exercise. We ask for the representation of the geometrical figure (the Riemann surface) so described.

Text appeared also in The Letter of THE IRIAN on July 3rd, 2007.

© Bukonzo Press, Paris, 2008.