Let ( X , ϕ ) and ( Y , σ ) be metric spaces, and let f , f 1 , f 2 , … bijective function with inverse functions g , g 1 , g 2 , … f n → f pointwise for n → ∞.And all involved functions are continuous. Does it hold that g n → g pointwise for n → ∞?

Question

Let   (  X  ,  ϕ  ) and   (  Y  ,  σ  ) be metric spaces, and let  f  ,      f    1    ,      f    2    ,  … bijective function with inverse functions   g  ,      g    1    ,      g    2    ,  …      f    n    →  f  pointwise for   n  →  ∞.And all involved functions are continuous. Does it hold that       g    n    →  g pointwise for   n  →  ∞?

Bruno Hughes · Accepted Answer

Partial answer:If   y  =  f  (  x  ) and       y    n    =      f    n    (  x  ) then      |    g  (  y  )  −      g    n    (  y  )      |    =      |    x  −      g    n    (    y  −      y    n    +      y    n      )      |    =      |        g    n    (      y    n    )  −      g    n    (      y    n    +      e    n    )      |  where       e    n    =  y  −      y    n    →  0.Therefore if   {      g    n    } is uniformly equicontinuous the answer is yes:      |    g  (  y  )  −      g    n    (  y  )      |    ≤  σ  (      e    n    )where   σ is the modulus of continuity of the family   {      g    n    }.

Let ( X , ϕ ) and ( Y , σ ) be

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