Prove that $f(\mathbb R) = \mathbb R$. Partially solved.

Question

Let $f:\mathbb R\to \mathbb R$ and $f(f(x)) = x + f(x)$, $x\in\mathbb R$

Prove that:
a) $f$ is injective

b) $f(0) = 0$

c) $f(\mathbb R) = \mathbb R$

My solution:

a) Let $x_1, x_2 \in \mathbb R$ and $f(x_1) = f(x_2)$ then

$f(f(x_1)) = f(f(x_2))$

$x_1 + f(x_1) = x_2 + f(x_2)$

$x_1 = x_2$ therefore $f$ is injective.

b) Let $x=0$ then $f(f(x)) = x + f(x)$

$f(f(0)) = 0 + f(0)$

$f(f(0)) = f(0)$ but $f$ is injective therefore

$f(0) = 0$

c) Here is the point where I get stuck. How do I solve this?

I think I have to begin saying, let $y\in\mathbb R$ and prove that there is a $x\in\mathbb R$ for which $f(x) = y$.

Answer 1

The solution (given by my teacher).

Let $y\in \mathbb R$ I'll prove that there is a $x \in \mathbb R$ for which $f(x) = y$

$f(x) = y$

$f(f(x)) = f(y)$

$x + f(x) = f(y)$

$x + y = f(y)$

$x = f(y) - y$

therefore $f(\mathbb R) = \mathbb R$

Answer 2

It works if you assume $f(x)$ is continuous.

Claim: Suppose $f(x)$ is continuous and $f(f(x)) = x + f(x)$ for all $x \in \mathbb{R}$. Then $f(\mathbb{R})=\mathbb{R}$.

Proof: We already know that $f(0)=0$ and $f(x)$ is injective. It follows that $f(1)\neq 0$.

-Case 1: Suppose $f(1)>0$. Then (by the intermediate value theorem) we must have: \begin{align} &f(x) > 0 \: \: \mbox{ if $x >0$} \\ &f(x) < 0 \: \: \mbox{ if $x<0$} \end{align}

For all positive integers $n$ we have: \begin{align*} &f(f(n)) = n + f(n) \geq n\\ &f(f(-n)) = -n + f(-n) \leq -n \end{align*} and so $f$ takes arbitrarily large values and arbitrarily small values. By the intermediate value theorem, it must take all values in $\mathbb{R}$.

-Case 2: Suppose $f(1)<0$. Then we must have:

\begin{align} &f(x) < 0 \quad \mbox{ whenever $x >0$} \\ &f(x) > 0 \quad \mbox{ whenever $x<0$} \end{align}

Suppose there is a finite constant $-M$ such that $f(x) \in [-M,0]$ for all $x\geq 0$ (we reach a contradiction). Then the infinite sequence $\{f(n)\}_{n=1}^{\infty}$ is in the compact interval $[-M,0]$, so the Bolzano-Wierstrass theorem ensures there is a subsequence $n_k$ such that $n_k\rightarrow\infty$ and $f(n_k) \rightarrow x^*$ for some $x^* \in [-M,0]$. But for all $k$ we have: $$ f(f(n_k)) = n_k + f(n_k) $$ and taking a limit as $k\rightarrow \infty$ gives $f(x^*) = \infty + x^*$, a contradiction. Thus, $f(x)$ takes arbitrarily small values over $x\geq 0$.

A similar argument shows $f(x)$ takes arbitrarily large values over $x \leq 0$. Hence, by continuity, $f(\mathbb{R}) = \mathbb{R}$.