Teorema di Darboux

Disambiguazione – Se stai cercando il teorema nella geometria simplettica, vedi Teorema di Darboux (geometria).

Il teorema di Darboux è un teorema dell'analisi matematica che prende il nome da Jean Gaston Darboux. Esso afferma che tutte le funzioni che risultano dalla derivazione di altre funzioni presentano la proprietà del valore intermedio: l'immagine di un intervallo è ancora un intervallo.

È da notare che quando è differenziabile con derivata continua (cioè ) questo è implicitamente vero per il teorema dei valori intermedi, ma anche quando non è continua il teorema di Darboux pone forti limiti alle sue variazioni.

Teorema di Darboux

Sia una funzione continua a valori reali in , che sia differenziabile in . Allora soddisfa la proprietà del valore intermedio: per ogni compreso tra e , esiste qualche in tale per cui .

Dimostrazione

Senza perdita di generalità si può supporre che . Sia , allora , quindi sostituendo, si ha , e si vuole trovare uno zero di .

Siccome è una funzione continua in , per il teorema di Weierstrass possiede un massimo in , ma questo massimo non può trovarsi in , poiché , quindi in un intorno destro di e in modo del tutto simile non può trovarsi in , poiché , quindi in un intorno sinistro di . Pertanto il massimo deve stare in un punto compreso in tale che per il teorema di Fermat sui punti stazionari, da cui la tesi.

Voci correlate

Collegamenti esterni

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