
Continuity problem
Hello MHF, please help on this problem; (and by the way, we only did the continuity lesson )
We are given :
$\displaystyle f:[0,1]\rightarrow [0,1]$
$\displaystyle g:[0,1]\rightarrow [0,1]$
both$\displaystyle f$ and $\displaystyle g$ are continuious on$\displaystyle [0,1]$
$\displaystyle f(g(x)) = g(f(x))$
If $\displaystyle f(a) = a $ then $\displaystyle f(g(a)) = g(a)$
Show that $\displaystyle \exists \alpha \in [0,1] f(\alpha) = g(\alpha)$
P.S : use proof by contradiction

Re: 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

Re: Continuity problem
Please I need to solve it before Monday morning.

Re: 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.

Re: Continuity problem
To start you off, suppose without loss of generality
$\displaystyle f(x)>g(x) \quad \text{ for an open interval in } [0,1] $
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.

1 Attachment(s)
Re: 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?