Let ( X , &#x03D5;<!-- ϕ --> ) and ( Y , &#x03C3;<!-- σ --> ) be

Davon Irwin

Davon Irwin

Answered question

2022-06-18

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 ?

Answer & Explanation

Bruno Hughes

Bruno Hughes

Beginner2022-06-19Added 24 answers

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 }.

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