Limes et continvitas · Topos Uranos

Svmmarivm

Haec lectio limites ad primam eorvm conseqventiam strvctvralem perdvcit, continvitatem scilicet, qvam lectio qvae seqvitvr ad omnes fvnctiones elementares extendit. Fvnctio in pvncto continva est cvm valor eivs ibi is est qvem valores propinqvi nvntiant; cvm limitibvs lectionvm svperiorvm haec notio scribitvr vt solita provocatio et responsio, vna tantvm differentia: ipsvm pvnctvm iam non exclvditvr. Ex qvo seqvitvr continvitatem in pvncto accvmvlationis eo redire vt limes exsistat et valori aeqvalis sit, et fere omnia eivs theoremata ex algebra limitvm sine nova demonstratione hereditate accipi. Discontinvitates in evitabiles, saltvs et essentiales classificantvr, et evitabiles extensione per continvitatem reparantvr. Vnicvm argvmentvm novvm est compositionis, vbi continvitas plvs qvam limes efficit: hypothesim tollit qvam theorema de limite compositionis postvlabat, et limitem intra fvnctionem continvam inferre sinit.

Proposita discendi

Continvitatem in pvncto, ex vna parte et in intervallo definire, eamqve cvm limite conferre.
Discontinvitates in evitabiles, saltvs et essentiales classificare, et fvnctionem cvm discontinvitate evitabili per continvitatem extendere.
Continvitatem svmmarvm, prodvctorvm, qvotientivm, radicvm et compositionvm fvnctionvm continvarvm demonstrare, et inde eam polynomiorvm, fvnctionvm rationalivm et fvnctionvm trigonometricarvm dedvcere.
Limites compvtare eos intra fvnctionem continvam inferendo, et fvnctiones in vno tantvm pvncto avt in nvllo continvas investigare.

Definitio

In lectione de limite fvnctionvm vnivs variabilis realis definitio pvnctvm $a$ de indvstria exclvdebat: limes describit qvid prope $a$ fiat, non in $a$ , et fieri potest vt fvnctio ibi definita non sit. Continvitas vtrvmqve confert: qvid prope fiat et qvid in ipso pvncto fiat.

DefinitioFvnctio in pvncto continva

Sint $f$ fvnctio in copia $A$ definita et $a$ elementvm copiae $A$ . Dicitvr $f$ esse continva in $a$ , si pro omni $\varepsilon > 0$ exsistit $\delta > 0$ tale vt omne $x$ copiae $A$ qvod condicioni $\abs{x - a} < \delta$ satisfacit etiam condicioni $\abs{f(x) - f(a)} < \varepsilon$ satisfaciat:

(\forall \varepsilon > 0)(\exists \delta > 0)(\forall x \in A)\left(\abs{x - a} < \delta \rightarrow \abs{f(x) - f(a)} < \varepsilon\right)

Si $f$ in $a$ continva non est, dicitvr esse discontinva in $a$ . Si in omnibvs pvnctis copiae $S$ in $A$ contentae continva est, dicitvr continva in $S$ ; et si in toto dominio svo continva est, dicitvr simpliciter continva.

Definitio a definitione limitis tribvs rebvs differt. Nvmervs ad qvem valores accedere debent non est qvodlibet $L$ , sed $f(a)$ ; qvam ob rem $a$ ad dominivm pertinere debet. Responsio $\abs{x - a} < \delta$ non exclvdit $x = a$ , qvamqvam id inclvdere nihil constat, qvia $\abs{f(a) - f(a)} = 0$ . Neqve postvlatvr vt $a$ pvnctvm accvmvlationis copiae $A$ sit. Negatio obtinetvr vt in lectione de limite: $f$ in $a$ discontinva est si exsistit $\varepsilon > 0$ tale vt, pro omni $\delta > 0$ , sit $x$ copiae $A$ cvm $\abs{x - a} < \delta$ et $\abs{f(x) - f(a)} \geq \varepsilon$ .

PropositioContinvitas et limes

Sint $f$ fvnctio in $A$ definita et $a$ elementvm copiae $A$ . Si $a$ pvnctvm accvmvlationis copiae $A$ est, tvm $f$ in $a$ continva est si et solvm si $\lim_{x \to a} f(x) = f(a)$ . Si $a$ pvnctvm solitarivm copiae $A$ est, tvm $f$ in $a$ continva est. Ambae definitiones idem postvlant praeterqvam in pvncto $a$ , vbi continvitas sponte valet. In figvra fascia valorem $f(a)$ circvmdat, qvi nvnc pvnctvm plenvm delineationis est; in fine, pvnctvm solitarivm dominii.

Demonstratio

$0 < \abs{x - a} < \delta \rightarrow \abs{x - a} < \delta \rightarrow \resaltar{\abs{f(x) - f(a)} < \varepsilon}$
Si $f$ in $a$ continva est, responsio $\delta$ ad qvamqve provocationem $\varepsilon$ nominatim valet pro iis $x$ qvae condicioni $0 < \abs{x - a} < \delta$ satisfacivnt: responsio est pro limite, cvm $L = f(a)$ .
$x = a \rightarrow \abs{f(a) - f(a)} = \resaltar{0} < \varepsilon$
Vicissim, si limes est $f(a)$ , responsio limitis ea $x$ complectitvr qvae a $a$ differvnt, et in $x = a$ condicio semper valet: responsio est pro continvitate.
$\abs{x - a} < \delta \wedge x \in A \rightarrow x = a \rightarrow \abs{f(x) - f(a)} = 0 < \varepsilon$
Si $a$ pvnctvm solitarivm est, exsistit $\delta > 0$ tale vt vnicvm elementvm copiae $A$ cvm $\abs{x - a} < \delta$ sit ipsvm $a$ . Illvd $\delta$ omnibvs provocationibvs simvl respondet.

Ergo in pvncto accvmvlationis dominii continvitas tribvs rebvs constat: vt $f$ in $a$ definita sit, vt limitem in $a$ habeat, et vt is limes sit $f(a)$ . Qvaeqve separatim deficere potest, et inde classificatio discontinvitatvm oritvr. Exempli gratia, fvnctio qvae valet $1$ in omni $x \neq 0$ et $0$ in $x = 0$ , ex exemplo contrario compositionis in lectione de algebra limitvm, limitem $1$ in $0$ habet, a valore svo diversvm: in $0$ discontinva est.

In reliqva parte hvivs lectionis

Continvitas lateralis et continvitas in intervallo
Classificatio discontinvitatvm
Operationes cvm fvnctionibvs continvis
Compositio
Proprietates locales
Problemata solvta
Dvo frvsta qvae conivngvntvr
Tres discontinvitates diversae
Oscillatio attenvata et altera non attenvata
Limites intra fvnctiones continvas
Fvnctio in vno tantvm pvncto continva
Problemata proposita
Tria frvsta
Valor absolvtvs fvnctionis continvae
Maximvm dvarvm fvnctionvm continvarvm
Fvnctio additiva
Fvnctio Thomaeana