Uniform continuity of composite function

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Discussion Overview

The discussion revolves around the uniform continuity of the composite function fg(x), given that both functions f and g are uniformly continuous. Participants are examining the proof provided and its validity, focusing on the conditions under which uniform continuity holds.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant presents a proof claiming that if f and g are uniformly continuous, then fg is also uniformly continuous.
  • Another participant questions the proof, suggesting that the domains of f and g need to be specified for the statement to hold true.
  • A participant expresses uncertainty about the correctness of the proof and seeks validation.
  • One participant confirms the proof's correctness, but this is challenged by the need for clarity on the domains of f and g.
  • A later reply clarifies that the intention was to consider f and g as uniformly continuous for all x, y in R.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the validity of the proof. There is disagreement regarding the necessity of specifying the domains of f and g for the proof to be valid.

Contextual Notes

The discussion highlights the importance of domain specifications in the context of uniform continuity, which remains unresolved in the proof provided.

estro
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I'll be very thankful is someone will tell me where I'm wrong.

We know:
1) f is uniform continuous.
2) g is uniform continuous.

We want to prove:
fg(x) is uniform continuous.

proof:
from 1 we know -> for every |a-b|<d_0 exists |f(a)-f(b)|<e
from 2 we know -> for every |x-y|<d exists |g(x)-g(y)|<d_0
let a=g(x) and b=g(y) then
for every x,y |x-y|<d exists |fg(x)-fg(y)|<e.

Sorry for my poor formulation, English is not my mother tongue.
 
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What's wrong with it?
 
I'll be more then happy if this is right...
I'm just not sure.
 
Yes, it's correct.
 
Estro:

You need to be careful to specify where (in their respective domains) f and g
are uniformly continuous. Otherwise your statement is not true. I think Zhentil
assumed f,g were everywhere unif. continuous.
 
Oh, thanks for the remark.
I indeed intended for every x,y in R.
 

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