Functie Darboux

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Lavinia
Euclid
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Joined: Mon Mar 24, 2008 10:54 pm
Location: Timisoara

Functie Darboux

Post by Lavinia »

Aratati ca nu exista functii cu proprietatea lui Darboux a.i. \( f(f(x))=\cos^2 x, \forall x \in {\mathbb R} \).
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c.adryan
Euclid
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Joined: Fri Feb 29, 2008 12:22 pm

Post by c.adryan »

Fie \( g:[0,1]\rightarrow[0,1],\ g(x)=f(x) \) pt orice \( x\in [0,1] \).
\( g(g(x))=cos^2x \) injectiva \( \Rightarrow \) g injectiva.
g are P.D. \( \Rightarrow \) g strict monotona.
g(g(x)) strict descrescatoare ceea ce e o contradictie.
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