Proving the Existence of f(x)=x: A Continuos Function Problem | [0,1]

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Homework Help Overview

The discussion revolves around proving the existence of a value \( x \) in the interval \([0,1]\) such that a continuous function \( f(x) \) satisfies \( f(x) = x \), given the conditions \( f(0) = 1 \) and \( f(1) = 0 \). The problem is situated within the context of continuous functions and their properties over a closed interval.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the implications of continuity and the behavior of functions at the endpoints of the interval. There are attempts to relate the problem to simpler cases, such as finding a root for a different continuous function \( g \). Questions arise about the necessity of a function crossing the x-axis based on its values at specific points.

Discussion Status

The discussion is ongoing, with participants seeking clarification and exploring various aspects of the problem. Some have proposed defining a new function \( G(x) = f(x) - x \) and raised questions about its continuity and the significance of its values at the endpoints. There is a recognition of hints provided by other participants that may relate to the original problem.

Contextual Notes

Participants are navigating the implications of continuity and the conditions set by the function values at the endpoints. There is an emphasis on understanding the relationship between the function and its intersection with the line \( y = x \). Some participants express confusion and seek further explanation of the concepts involved.

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Homework Statement



Suppose that f is ais continuos function defined on [0,1] with f(0)=1 and f(1)=0. show that there is a value of x that in [0,1] such that f(x)=x. Thank You.
 
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This is slightly simpler:
Let's say I have a function [itex]g:[0,1] \rightarrow \Re[/itex], which is continuous, with [itex]g(0)=1[/itex] and [itex]g(1)=-1[/itex] can you show that there is an [itex]x \in [0,1][/itex] so that [itex]g(x)=0[/itex]?
 
Im Still Not Following. Can You Explain It A Little Bit Better To Me. Thank You For Your Post By The Way.
 
If a function that's continuous is negative at one point, and positive at another point, does it necesarily cross the x-axis (i.e. is zero somewhere in between)?

That's what he's driving at, but puts it in terms that are more obviously applicable to the problem at hand
 
Define the function G(x)=f(x)-x. Now four questions. i) is G continuous? ii) What are G(0) and G(1)? iii) What does it mean if G(x)=0 in terms of f? iv) Might this have something to do with the NateTG's and Office_Shredder's hints?
 

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