Problem 3

Let

\mathbb{R}_+

denote the set of positive real numbers. Determine all functions

f:\mathbb{R}_+\to\mathbb{R}

such that for all

x,y,z\in\mathbb{R}_+

|x-y|<|y-z|\ \text{if and only if}\ |f(x)-f(y)|<|f(y)-f(z)|.

Step 5 of 5: Normalize and pin down

f

by density

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f(1)=1,f(2)=2 \implies f\!\left(\tfrac{n}{2^k}\right)=\tfrac{n}{2^k}\ \forall n\in\mathbb{Z}_{>0},k\ge0 \implies f(x)=x\ \forall x\in\mathbb{R}_+

Detailed analysis

Because $f$ is injective, $f(2)\ne f(1)$ . Define $g(x)=1+(f(x)-f(1))/(f(2)-f(1))$ . Affine changes of the range preserve the condition, so $g$ also satisfies it and has $g(1)=1,g(2)=2$ . Applying the midpoint property repeatedly gives $g(n/2^k)=n/2^k$ for every positive integer $n$ and $k\ge0$ . These values are dense in $\mathbb R_+$ , so monotonicity forces $g(x)=x$ ; undoing the normalization gives the original form $f(x)=ax+b$ .