4:50 AM
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Suppose $f(x)$ and $g(x)$ are continuous functions on $[a,b]$ with $f$ monotone increasing. Assume there exists a sequence $x_n \in [a, b]$ such that for all $n \in \mathbb{N}$ , $g(x_n) = f(x_{n+1})$. Show that there exists $x_0 \in [a,b]$ such that $g(x_0) = f(x_0)$. Just wondering if this que...

It is answered in the linked question, just to explain why the function you suggested are not a counterexample @user136266.
You suggested $g(x)=x+1$ and $f(x)=x$.
In this case, the equality $g(x_n)=f(x_{n+1})$ is fulfilled if and only if $x_{n+1}=x_n$.
So if you star with arbitrary $x_1$, the rest of the sequence is uniquely determined:
$x_2=x_1+1$, $x_3=x_2+1=x_1+2$, ..., $x_n=x_1+(n-1)$.
This would indeed be a counterexample showing that the claim from the question if not true for function $f\colon \mathbb R\to\mathbb R$.
But the question is about a functions defined in the closed interval $[a,b]$. The sequence described above is unbounded, it cannot stay in this interval.
So this is not a counterexample for the claim from your question (which is about functions $[a,b]\to\mathbb R$.
It is worth mentioning that in the solution I have suggested in an answer to another question I have used that the sequence $(x_n)$ is bounded.
(If my solution did not use some property of $[a,b]$, which is not true for $\mathbb R$, we would know that there is a mistake in that proof. Since such solution would be attempt to prove the claim for functions defined on $\mathbb R$, too. And you have find a counterexample to such claim.)
I hope this cleared things at least a little.

5:19 AM
@MartinSleziak Hi.

Hi!
Good afternoon, I guess...
(It would be "Good morning!" here.)

Its good morning, especially for me (means that I woke up now)! 10:50.

7:20 here
I thought that there are only few time zones where the difference from CET is of by some number of hours and 30 minutes.
OTOH I never really needed to follow time zones that closely.

@MartinSleziak Are you a math professor?

I think the best translation to English would be teaching assistant or assistant professor?

5:23 AM
[Anyways, I think you should run for moderator election next time.]

I do work at university. In Slovakia.

Ok :)

I am not sure what would be correct equivalent in the American university system. (I think British system is similar.)
I am afraid I would not be a good moderator. It requires people skills, even if you do not meet people personally, only over the internet. And I usually have periods of inactivity on MSE, which would not be a good thing for a mod.

5:35 AM
@MartinSleziak Do you know of any quantities related to $\cos \left(\frac{B-C}{2}\right) + \cos \left(\frac{C-A}{2}\right) + \cos \left(\frac{A-B}{2}\right)$ in a triangle. Or where could I hope to find them?

@Sawarnik Not that good in geometry, definitely I do not know anything from the top of my head.
If you do not find anything after some research, it is worth asking on main.

Ok :)

I have also reposted your query in geometry chatroom.

Thanks! :)

I am not sure whether it will help. (Probably not many users look into that room) But at least it will keep the room from getting frozen.

3 hours later…
8:34 AM
@Martin Sleziak can you provide me with two functions that fulfil this property, so that i can better understand the formula. SO far, i still can't get an intuitive feel of what the question is trying to tell me — user136266 1 min ago
You can basically choose any two surjections $[0,1]\to[0,1]$.
For example $f(x)=x$ and $g(x)=x^2$.
Maybe you could draw a pictures with the iterations, something like this but for these two functions.
The picture was taken from this post‌​.
Or this is something similar:
It is taken from here.
Or this one:
I was not able to find a picture which would have exactly the functions I suggested, but these could give you an idea.

9:09 AM
This is the picture I was going for:
It is the situation from your problem for functions $x$ and $\sqrt x$ (on interval $[0,1]$.)

5 hours later…
2:20 PM
@MartinSleziak Hi. Can you look into this question? math.stackexchange.com/questions/770069/…
Daniel says that there is no accepted definition, but Bartle's is better. As far as I understood his words.

I agree that Bartle's definition is more general.
And I would even dare to say more usual.

@MartinSleziak Ok :) So if someone ask that if that limit exist, the accepted answer should be, it does?

2:41 PM
@Sawarnik Yes, i would say that the limit exists.

=) Thanks, I was thinking I was making a blunder somewhere. No prob now :)