Problem 5

Determine all functions

f:\mathbb R\to\mathbb R

such that (1)

f

is strictly increasing, and (2)

f(x)+g(x)=2x

for every real

x

, where

g

is the composition inverse of

f

Step 4 of 5: Monotonicity bounds every multiple of a difference

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x>y\Rightarrow x+n(a-b)>y,\qquad x-n(a-b)>y,\quad a=f(x)-x,\ b=f(y)-y

Detailed analysis

For $x>y$ , the increasing maps $f_n$ and $g_n$ preserve order. Using $f_n(x)=x+na$ gives the first inequality. Using $g_n(x)=x-na$ from the previous step gives the second. Thus both $x-y+n(a-b)$ and $x-y-n(a-b)$ are positive, so $|n(a-b)|<x-y$ for every positive integer $n$ .