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Three functions and a binary operation

Source: Romanian District Olympiad 2023 12.4

March 11, 2023

real analysisabstract algebrafunction

Problem Statement

Consider the functions

f,g,h:\mathbb{R}_{\geqslant 0}\to\mathbb{R}_{\geqslant 0}

and the binary operation

*:\mathbb{R}_{\geqslant 0}\times \mathbb{R}_{\geqslant 0}\to \mathbb{R}_{\geqslant 0}

defined as

x*y=f(x)+g(y)+h(x)\cdot|x-y|,

for all

x,y\in\mathbb{R}_{\geqslant 0}

. Suppose that

(\mathbb{R}_{\geqslant 0},*)

is a commutative monoid. Determine the functions

f,g,h

.

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