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If I have function from $R^3$ to $R$ satisfying

$f(x_1,x_2,x_3)+f(y_1,y_2,y_3) = f(x_1+y_1,x_1+y_2,x_3+y_3)$

is it necessarily linear?

$f(z_1,z_2,z_3) = \lambda _1 z_1+\lambda _2 z_2+\lambda _3 z_3$

Wasn't sure if this was a direct consequence of Cauchy's theorem or not.

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Without any additional assumptions like continuity or boundedness, you can't prove linearity. If $a$, $b$ and $c$ are additive functions, then the function defined below satisfies the functional equation: $$f(x,y,z)=a(x)+b(y)+c(z)$$

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