find all continuous function $f:\mathbb R^+\to\mathbb R^+$ satisfy: $f(x)=f(\frac{x+1}{x+2})$

Question

1. Find all continuous function $f:\mathbb R^+\to\mathbb R^+$ satisfy: $f(x)=f(\frac{x+1}{x+2})$
2. Find all continuous function $f:\mathbb R\to\mathbb R$ satisfy $\forall a<b, \exists c \in (a,b): f(c)\ge \max\{f(a),f(b)\}$

Answer 1

1. Let $x_1=x$ and $x_{n+1}=\frac{x_n+1}{x_n+2}$. What can you say about $\lim_n x_n$? What about $f(x_n)$?

2. Are you sure $c \in [a,b]$? It is easy to see that $f(a) \geq \max\{f(a),f(b)\}$ or $f(b) \geq \max\{f(a),f(b)\}$.

Updated If $c \in (a,b)$ the problem is harder.

Assume that you can find $x_1,x_2$ so that $f(x_1) < f(x_2)$. I will cover the case $x_1<x_2$, the other one is similar.

Now, you can find a smallest $x_3 \in [x_1, x_2]$ such that $f(x_3)=f(x_2)$ (WHY?). Apply the condition from the problem to $a=x_1, b=x_3$, and the by IVT you can find a $y$ between $x_1$ and $c$ so that $f(y)=f(x_3)$. Combine this with the fact that $x_3$ is smallest, reason, and you get the answer.

I probably provided too many details for a Homework problem ...