f , g : R → R continuous functions such that f ( g ( x ) ) = x. Prove that g ( f ( x ) ) = x.Since f is surjective, we basically have to prove that f is injective. For g is clearly, but how can I prove that f is injective ? Please, help me!

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f  ,  g  :      R    →      R   continuous functions such that   f  (  g  (  x  )  )  =  x. Prove that   g  (  f  (  x  )  )  =  x.Since   f is surjective, we basically have to prove that   f is injective. For   g is clearly, but how can I prove that   f is injective ? Please, help me!

Miguidi4y · Accepted Answer

As   g is continuous and injective, it is strictly monotonicif   ℓ  :=      lim          x      →      ∞        g  (  x  )  ∈            R      , then       lim          x      →      ∞        f  (  g  (  x  )  )  =  f  (  ℓ  ), contradicting   f  (  g  (  x  )  )  =  x. Hence       lim          x      →      ∞        g  (  x  )  =  ±  ∞Same result with       lim          x      →      −      ∞        g  (  x  )conclude that   g is surjective.If   y  =  g  (  x  ) then   f  (  y  )  =  f  (  g  (  x  )  )  =  x, hence   g  (  f  (  y  )  )  =  g  (  x  )  =  y

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