Hello MHF, please help on this problem; (and by the way, we only did the continuity lesson )

We are given :

both and are continuious on

If then

Show that

P.S : use proof by contradiction

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- September 28th 2013, 10:13 AMOrpheusContinuity problem
Hello MHF, please help on this problem; (and by the way, we only did the continuity lesson )

We are given :

both and are continuious on

If then

Show that

P.S : use proof by contradiction - September 28th 2013, 10:35 AMOrpheusRe: Continuity problem
I tried to do this

let h(x) = f(x) - g(x)

for every x in [0,1] h(x)≠0

then for every x in [0,1] h(x) > 0 ( or < 0)

but after this i could't find a contradiction - September 28th 2013, 12:52 PMOrpheusRe: Continuity problem
Please I need to solve it before Monday morning.

- September 28th 2013, 06:38 PMjohngRe: Continuity problem
Hi,

Since you need this by Monday morning, I can only assume it's either an exam question or homework for credit. So here are some hints:

Let a be a fixed point of f (f(a)=a) -- you need to prove why a exists. Now consider the sequence x_{1}= a and x_{n+1}= g(x_{n}).

By the way, the above leads to a direct proof, not a proof by contradiction. - September 28th 2013, 10:01 PMchen09Re: Continuity problem
To start you off, suppose without loss of generality

You'll get a contradiction along the way.

Note: Using the auxiliary function h you defined, could be used for a direct proof, but at that point you guys are assumed to know Intermediate Value Theorem, etc.

Disclaimer: If my line of thought is wrong or it has missing holes, please correct and/or inform me. - October 1st 2013, 06:35 AMjohngRe: Continuity problem
Hi again,

If you're still interested after your deadline, here's a complete solution:

Attachment 29357

Chen, I'm interested in seeing a proof along the lines suggested by your post. Could you provide such?