Functional equation II

Algebra Level 3

Determine the number of real-valued continuous functions f f that satisfy f ( f ( x ) ) = − x f\left( f\left( x \right) \right) =-x for all x ∈ R x\in\mathbb{R} .


The answer is 0.

Félix de Montemar by Félix de Montemar , 23, Spain

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1 solution

It is easy to prove that f f is injective, because f ( x ) = f ( y ) ⇒ f ( f ( x ) ) = f ( f ( y ) ) ⇒ − x = − y ⇒ x = y f\left( x \right) =f\left( y \right) \Rightarrow f\left( f\left( x \right) \right) =f\left( f\left( y \right) \right) \Rightarrow -x=-y\Rightarrow x=y Since f f is continuous, it must be monotone, whether f is increasing or decreasing, f ( f ( x ) ) f\left( f\left( x \right) \right) is increasing, but g ( x ) = − x g\left( x \right) = -x is a decreasing function, and this is a contradiction

1 Helpful 0 Interesting 0 Brilliant 0 Confused

But an injective function does not have to be monotone. In fact, such functions do exist. See http://math.stackexchange.com/questions/312385/continuous-function-f-mathbbr-to-mathbbr-such-that-ffx-x .

Jon Haussmann - 5 years, 2 months ago

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Yes, you are right. My apologies!

Félix de Montemar - 5 years, 2 months ago

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