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  • existe t-il une fonction f:RR f :{\mathbb R}\longrightarrow{\mathbb R} telle que : f(f(x))+xf(x)=1f(f(x))+xf(x)=1 pour tout xRx\in {\mathbb R} ?

  • NoNo Because P(0)P(0) f(f(0))=1\Rightarrow f(f(0))=1
    P(f(0))f(0)+f(1)=1P(f(0))\Rightarrow f(0)+f(1)=1
    P(1)f(f(1))+f(1)=1P(1) \Rightarrow f(f(1))+f(1)=1
    So f(f(1)=f(0) f(f(1)=f(0) So f(f(f(1)))=1f(f(f(1)))=1
    P(f(1))f(1)f(f(1)=0P(f(1)) \Rightarrow f(1)*f(f(1)=0
    If f(1)=0f(1)=0 Then f(0)=1f(0)=1 Then f(1)=f(f(0))=1f(1)=f(f(0))=1 Absurde!
    If f(f(1))=0 f(f(1))=0 then f(1)=1 f(1)=1 Then f(f(1)=f(1)=1f(f(1)=f(1)=1 Absurde!
    So there is no function that satisfy the given conditions

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