For function $f:[0,1]\rightarrow \Bbb R$ we know : $f(x+y)\geq f(x)+f(y)$.Prove that $f(x)\leq 2x$

Question

Function $f:[0,1]\rightarrow \Bbb R$ holds in the following conditions :
1) $f(1)=1$
2) $\forall x \in [0,1]: f(x)\geq 0$
3) $\forall x,y,x+y \in [0,1]: f(x+y)\geq f(x)+f(y)$.
Prove that $\forall x \in [0,1] : f(x)\leq 2x$

I tested values $0,\frac12$ in the given formula and got the following results:
$$f(0)=0 , f(\frac12)\leq \frac12,f(x)\leq \frac{f(2x)}{2}$$
Please help complete the proof.

Answer 1

Case 1: $x ≥ 1/2$.

Let $x \in [0,1]$, and set $y = 1 - x$. Then $1 = f(x+y)≥ f(x)+f(y) ≥ f(x)$. This tells us that $f$ is bounded above by $1$. Hence to show that $f(x) ≤ 2x$ for all $x \in [0,1]$, it now suffices to show that it holds for $x < 1/2$. (If $x ≥ 1/2$, then $f(x) ≤ 1 ≤ 2x$ so we are done.)

Case 2: $1/4 ≤ x < 1/2$.

We also have, as you discovered, $$f(1/2) = \frac{f(1/2)+f(1/2)}{2} \leq \frac{f(1)}{2} = \frac{1}{2}$$ Let $x \in [0,1/2)$. Let $y = 1/2-x$. Then $1/2 ≥ f(x+y) ≥ f(x) + f(y) ≥ f(x)$. So now, by the same argument as case 1, we have $f(x) ≤ 2x$ for $x ≥ 1/4$.

Case 3: $1/8 ≤ x < 1/4$. $$f(1/4) = \frac{f(1/4)+f(1/4)}{2} \leq \frac{f(1/2)}{2} ≤ \frac{1}{4}$$

Can you see the pattern? Given any $x \in (0,1]$, a "finite descent" tells us that $f(x) ≤ 2x$. As for $x = 0$, we see that $0≤ f(0) ≤ f(1+0) - f(1) = 1-1 = 0$.

Hence we are done!

Answer 2

By induction, (3) easily generalizes to

$$ \forall x_1,x_2,\ldots x_n, x_1+x_2+\ldots+x_n\in [0,1], f\bigg(\sum_{k=1}^n x_k \bigg) \geq \sum_{k=1}^n f(x_k) \tag{4} $$

Putting $x_1=x_2=\ldots=x_n=\frac{1}{n}$ in (4) and using (1), we deduce :

$$ f(\frac{1}{n}) \leq \frac{1}{n} \tag{5} $$

Now, let $x\in [0,1]$ with $x\neq 0$. There is an integer $k$ such that $\frac{1}{2^{k+1}} \lt x \leq \frac{1}{2^k}$. Using (3) with $y=\frac{1}{2^k}-x$, we deduce $f(x)\leq f(\frac{1}{2^k})$. But $f(\frac{1}{2^k}) \leq \frac{1}{2^k}$ by (5), and hence $f(x) \leq \frac{1}{2^k} \leq 2x$ as wished.

All that's left is the case $x=0$. But this is easy : taking $x_1=x_2=\ldots=x_n=0$ in (4) and using (1), we have $f(0) \leq \frac{1}{n}$ for all $n$, whence $f(0)\leq 0$ by pasing to the limit, whence $f(0)=0$ by (2).

Answer 3

$f(0)=0$ so $f(0)\le2\times0$

$f(1)=1$ so $f(1)\le2\times1$

Take $x=y=\frac{1}{2}$, we get

$f(\frac{1}{2}) \le \frac{1}{2}f(1) = \frac{1}{2}$

Take $x=y=1/4$, we get

$f(\frac{1}{4}) \le \frac{1}{2}f(\frac{1}{2}) = \frac{1}{4}$

So, $f(\frac{1}{2^n}) \le \frac{1}{2^n}$

Also, $f$ is nondecreasing and it's range is $[0,1]$.

Thus, for $[0,1]$, $f(x)\le 2x$