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If I have an additive function which is [itex]f(x+y)=f(x)+f(y)[/itex],

the question is

how can we prove that if this function has a limit at each real number then there is a number a greater than zero and M greater than zero

such that

[itex]|f(x)|\leq M[/itex], for all [itex]x\in[-a,a][/itex],

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# The additive function is bounded

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