Calculus and analysis

Martin Sleziak

04:50

room topic changed to Calculus and real analysis: For questions about calculus and real analysis [calculus] [real-analysis]

For instructions how to render MathJax(TeX) in chat see this post on meta.

About the following question:

Q: Some Continuity Question

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.

Sawarnik

05:19

@MartinSleziak Hi.

Martin Sleziak

Hi!

Good afternoon, I guess...

(It would be "Good morning!" here.)

Sawarnik

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

Martin Sleziak

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.

Sawarnik

@MartinSleziak Are you a math professor?

Martin Sleziak

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

Sawarnik

05:23

[Anyways, I think you should run for moderator election next time.]

Martin Sleziak

I do work at university. In Slovakia.

Sawarnik

Ok :)

Martin Sleziak

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.

Sawarnik

05:35

@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?

Martin Sleziak

@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.

Sawarnik

Ok :)

Martin Sleziak

I have also reposted your query in geometry chatroom.

Sawarnik

Thanks! :)

Martin Sleziak

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…

Martin Sleziak

08:34

@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.