I would like to know what are the especifications of a functional equation that give us a power function as a solution.
For example, if f:R→R is continuous and monotonic, such that f(x)+f(y)=f(z) iif f(λx)+f(λy)=f(λz) for all λ>0, then f(x)=axb.
Does anyone know another functional equation that gives a power function as a solution?
Answer
For all x and y≠0, the only continuous solutions of the equation f(x)2=f(xy)f(x/y) is
f(x)=axb.
An alternative equation similar to
Cauchy's functional equation is
f(1)f(x y)=f(x)f(y) for all x,y.
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