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cauchy´s functional equation

Source: Romanian District Olympiad 2015, Grade X, Problem 4

September 25, 2018

functionalgebrafunctional equation

Problem Statement

Let

f: (0,\infty)\longrightarrow (0,\infty)

a non-constant function having the property that f\left( x^y\right) = \left( f(x)\right)^{f(y)}, \forall x,y>0. Show that

f(xy)=f(x)f(y)

and

f(x+y)=f(x)+f(y),

for all

x,y>0.