MathDB

Let

a\in \mathbb{R}

and

f_1(x),f_2(x),\ldots,f_n(x): \mathbb{R} \rightarrow \mathbb{R}

are the additive functions such that for every

x\in \mathbb{R}

we have

f_1(x)f_2(x) \cdots f_n(x) =ax^n

. Show that there exists

b\in \mathbb {R}

and

i\in {\{1,2,\ldots,n}\}

such that for every

x\in \mathbb{R}

we have

f_i(x)=bx

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