Charta functionis exemplaris,

Functio in arte mathematica est congruentia inter duas copias, quae ad quodque elementum primae copiae unum elementum secundae copiae destinat.[1] Prima copia dominium dicitur,  altera codominium. Si quoddam elementum primae copiae designat, variabilis independens est. Si functio est, quae mathematice per scribitur et significat " esse elementum codominii ad elementum dominii destinatum", variabilis dependens appellatur.

Si plura elementa sunt, quae ad elementum destinari possint, congruentia non est functio. Exempli gratia: sit sintque dominium et codominium copiae numerorum realium . Haec congruentia non est functio, quod elemento (velut 4) duo elementa (i. e. 2, -2) attribuuntur. Sin autem codominium est copia numerorum realium non-negativorum vel functio est velut , haec congruentia functio appellatur.

Functionem definire licet per formulam aut regulam aut tabulam, dum modo unum elementum codominii sit quod ad quodque elementum dominii destinetur.

Analysis est theoria functionum. Analysis numerorum realium est theoria functionum quarum dominium (et codominium) est ; analysis numerorum complexorum est analysis earum, quarum dominium est . G. H. Hardy dicit, "Illa notio, quantitatem variabilis ex quadam alia dependere, fortasse notio potissima in tota arte mathematica est."[2]

Si dominium est copia quantitatum binarum, sicut , functio duas variabiles independentes habet. Exempli gratia, . Quae functio par elementorum ad unum elementum codominii (quod est ) destinat, velut par ad . Functiones autem tres, quattuor pluresve variabiles independentes habere possunt.

Altera notatio functionum est notatio lambda, quae variabiles independentes post lambda litteram enumerat. Scribitur: quod eandem functionem atque describit. Forma sicut appellatur combinatoria.

Si ad quoddam elementum codominii aut nullum aut unum elementum dominii destinatur, functio appellatur functio iniectiva, aut functio unum elementum uni elemento attribuens. Si omne elementum codominii habet elementum (aut plura elementa ...) dominii quod ad destinatur, functio appellatur functio superiectiva. Functio quae simul iniectiva et superiectiva est, functio biiectiva appellatur.

Quaedam functio biiectiva habet functionem inversam , cuius dominium est codominium functionis cuiusque codominium dominium functionis . Si , tum est . Exempli gratia: functio habet functionem inversam . Formulam functionis inversae describere saepenumero haud facile est.

Compositio functionum est nova functio per quam elementum dominii primae functionis cum elemento codominii secundae functionis congruit. Si sunt functiones, et si dominium functionis est (aut continet) codominium functionis , scribi potest . Exempli gratia, sint . Nunc , et . Non sunt eaedem functiones: si , sed .

Copia omnium functionum invertibilium quarum dominium et codominium eadem copia est caterva appellatur. Idemfactor catervae est functio quae quodque elementum cum eodum elemento coniungit: ; operatio catervae est compositio.


  1. Behnke et al, p. 64.
  2. Hardy, p. 40.


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