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 5 of 5: Force the displacement to be constant and verify

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|n(a-b)|<x-y\forall n\in\mathbb Z_{>0}\Rightarrow a=b\Rightarrow f(x)=x+c

Detailed analysis

If $a\ne b$ , then $|n(a-b)|$ eventually exceeds the fixed number $x-y$ , contradiction. Thus $f(x)-x=f(y)-y$ for all $x,y$ , so $f(x)=x+c$ . Its inverse is $g(x)=x-c$ , and $f(x)+g(x)=2x$ ; every real constant $c$ therefore works.