Function well defined?
If the function f : R —> R is given by
f(x) := sin(cos(sin(cos(….(x))…)))
well-defined?
If so, does it have a derivative?
PS: I thought of that myself 🙂
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Schau Dir den Fixpunktsatz von Banach an.
Edit: Setze
und mit einem beliebigen Startwert x_0. Dann entspricht Deine Funktion f:
Die Frage ist nun, ob phi eine Kontraktion ist. Wenn ja, dann hätte sie einen eindeutigen Fixpunkt und der Grenzwert, also f(x_0), konvergierte dann für alle x_0 gegen diesen Fixpunkt.
Damit wären beide Deiner Fragen unmittelbar beantwortet.
Ja klar, dafür muss f kontrahierend sein. Aber worauf willst Du hinau?
Not only that, f must also be a self-map, which is the case here. Your sequence should always converge to the unique fixed point (no matter what you substitute for x).
I added something above.