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This questions comprises two independent parts.
(i) Let

g:\mathbb{R}\to\mathbb{R}

be continuous and such that

g(0)=0

and

g(x)g(-x)>0

for any

x > 0

. Find all solutions

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

to the functional equation

g(f(x+y))=g(f(x))+g(f(y)),\ x,y\in\mathbb{R}

(ii) Find all continuously differentiable functions

\phi : [a, \infty) \to \mathbb{R}

, where

a > 0

, that satisfies the equation

(\phi(x))^2=\int_a^x \left(\left|\phi(y)\right|\right)^2+\left(\left|\phi'(y)\right|\right)^2\mathrm{d}y -(x-a)^3,\ \forall x\geq a.

Problem Statement