Theorema valoris intermedii

Svmmarivm

Haec lectio primvm magnvm theorema de fvnctionibvs in intervallo continvis demonstrat: fvnctio continva a valore ad alivm transire non potest nisi omnes valores intermedios capiat. Primvm demonstratvr theorema Bolzani, qvod nihilvm praestat cvm fvnctio signvm mvtat, per axioma completvdinis; ex eo colligvntvr theorema valoris intermedii, imaginem intervalli intervallvm esse, et inversam fvnctionis continvae et stricte monotonae continvam esse, qvod derivata inversae reqvirit. Theorema non dicit vbi nihilvm sit, sed methodvs bisectionis eivs locvm qvalibet praecisione determinat. Lectio id adhibet ad exsistentiam radicvm aeqvationvm, ad polynomia gradvs imparis et ad pvncta fixa.

Proposita discendi

Theorema Bolzani per axioma completvdinis demonstrare et ex eo theorema valoris intermedii dedvcere.
Demonstrare imaginem intervalli per fvnctionem continvam intervallvm esse, et continvitatem inversae fvnctionis continvae et stricte monotonae dedvcere.
Theoremate vti ad exsistentiam solvtionvm aeqvationvm ostendendam, earvmqve locvm methodo bisectionis determinare.

Theorema Bolzani

Delineatio fvnctionis in intervallo continvae, secvndvm pictvram intvitivam, stilo non svblato dvcitvr; si infra axem incipit et svpra desinit, evm transire debet. Haec intvitio demonstrationem reqvirit, qvia ex nvmeris realibvs pendet: fvnctio $x^2 - 2$ in $0$ negativa est et in $2$ positiva, sed si soli nvmeri rationales considerarentvr, nvmqvam evanesceret, qvia $\sqrt{2}$ rationalis non est. Demonstratio igitvr axiomate completvdinis vtitvr, per svpremvm.

TheoremaTheorema Bolzani

Sit $f$ in $\icc{a}{b}$ continva, cvm $f(a) < 0 < f(b)$ . Tvm exstat $c$ in $\ioo{a}{b}$ ita vt $f(c) = 0$ . Idem valet si $f(a) > 0 > f(b)$ . Svmitvr svpremvm pvnctorvm in qvibvs fvnctio negativa est; conservatio signi ex lectione de limite et continvitate prohibet ne fvnctio in eo negativa avt positiva sit. Figvra copiam horvm pvnctorvm eivsqve svpremvm ostendit, deinde dvos casvs qvi fieri non possvnt.

Demonstratio

$S = \left\{ x \in \icc{a}{b} : f(x) < 0 \right\}, \qquad c = \resaltar{\sup S}$
Sit $S$ copia eorvm $x$ ex $\icc{a}{b}$ in qvibvs $f(x) < 0$ . Vacva non est, qvia $a$ continet, et $b$ margo svperior est; per axioma completvdinis svpremvm $c$ habet, cvm $a \leq c \leq b$ .
$f(c) < 0 \rightarrow c + \frac{\delta}{2} \in S, \qquad c + \frac{\delta}{2} > c$
Si $f(c) < 0$ , tvm $c \neq b$ , qvia $f(b) > 0$ . Per conservationem signi, $f$ in $\ico{c}{c + \delta}$ negativa est pro aliqvo $\delta > 0$ , qvod svmi potest cvm $c + \delta \leq b$ ; tvm $S$ pvncta maiora qvam $c$ continet, et $c$ margo svperior non esset.
$f(c) > 0 \rightarrow \sup S \leq c - \delta < c, \qquad \resaltar{f(c) = 0}$
Si $f(c) > 0$ , tvm $c \neq a$ . Per conservationem signi, $f$ in $\ioc{c - \delta}{c}$ positiva est pro aliqvo $\delta > 0$ , qvod svmi potest cvm $c - \delta \geq a$ , ita vt nvllvm pvnctvm eivs intervalli in $S$ sit; cvm neqve vlla maiora qvam $c$ sint, nvmervs $c - \delta$ margo svperior esset svpremo minor. Ergo $f(c) = 0$ , et $c$ neqve $a$ neqve $b$ est. Casvs alter ad primvm redvcitvr per $-f$ .