Does there exist a continuous function $f\colon \Bbb R\rightarrow \Bbb R$ such that $f(f(x))=-x$ for all $x\in\Bbb R$?

Question

Does there exist a continuous function

$f\colon \Bbb R\rightarrow \Bbb R$ such that $f(f(x))=-x$ for all $x\in\Bbb R$?

actually trying to disprove – user128956 Mar 05 '14 at 14:46 — user128956, Mar 05 '14 at 14:46

score 7 · Answer 1 · answered Mar 05 '14 at 14:47

7

Hint: If $f(f(x))=-x$ then $f$ is a bijection and because $f$ is continuous it must also be either order preserving or order reversing.

answered Mar 05 '14 at 14:47

Dan Rust

1

not getting.........elaborate please – user128956 Mar 05 '14 at 14:48
1

I think the hint should be enough for you to get to the answer. Write down the definition of an order preserving/reversing function and see what conclusions you can make from the above. – Dan Rust Mar 05 '14 at 14:53
but why is $f$ a bijection? – user2345215 Mar 05 '14 at 14:54
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$f\circ f$ is a bijection, and if $f$ was not a bijection then either $f\circ f$ would not be injective, or it would not be surjective - hence $f$ is bijective. – Dan Rust Mar 05 '14 at 14:55
$f\circ f\circ f\circ f$ is the identity, hence a bijection. Thus $f$ has to be a bijection, too. – Jyrki Lahtonen Mar 05 '14 at 14:55

1 Answers1