Continuously uniform function proof

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Homework Help Overview

The discussion revolves around the properties of a strict growing continuous function defined on the interval (0,1) and whether such a function must be continuously uniform. The original poster expresses uncertainty about how to approach the problem.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants discuss the potential use of counterexamples, specifically mentioning the function f(x) = -1 / (x - 1) and its implications for uniform continuity. Questions arise regarding the appropriate theorems to apply and the reasoning behind the strict growth condition.

Discussion Status

There is an ongoing exploration of the properties of the function in question, with some participants suggesting graphical analysis and links to external resources. The discussion reflects a mix of interpretations and approaches without reaching a consensus on the proof strategy.

Contextual Notes

Participants note the specific domain and range of the function, as well as the implications of strict growth on continuity. There is a mention of the difficulty in understanding the necessity of the strict growth condition in the context of the proof.

aeronautical
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Homework Statement



Let f : (0,1) → (0,1) be a strict growing continuous function. Does f have to be continuously uniform? Please note that its from (0,1) → (0,1) and NOT [0,1] → [0,1]. Please help me with the steps...I have no clue where to start...thanks...
 
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Consider the function [tex]f(x) = -1 / (x - 1)[/tex]. Prove that this is a counterexample.
 
Last edited:
Sorry but could you elaborate further? What theorem should I use?
 
Sorry, the function in my previous post should have been [tex]f(x) = -1 / (x - 1)[/tex]

You don't need to use any special theorem. Just try drawing a graph of this function, and see if you can prove that it is not uniformly continuous on (0,1).

If you find that difficult, think about the following easier example. Prove that 1/x is not uniformly continuous on (0,1).
 
The function I wrote down is the same function except reflected in the y axis, so that it is strictly increasing (a condition in the question), and moved to the right so that the infinity is at 1 instead of at 0.
 
I drew the function. It goes to plus minus infinity at x=1. So what does this tell me? That it is not continuously uniform?
 
Here is my plot...
 

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Last edited:
aeronautical said:
I drew the function. It goes to plus minus infinity at x=1. So what does this tell me? That it is not continuously uniform?

You will notice that it is the same as the function 1/x except moved and reflected. You posted a link to a website showing how to prove that 1/x in not uniformly continuous on (0,1), so you can use the same method with slight modifications to prove that this one is not uniformly continuous.
 
  • #10
However, I do not understand why the specify that (0,1) → (0,1) is strict growing continuous function. So this proof is by contradiction?
 

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