I would like to know why if it is not valid. Thanks in advance.
2026-04-09 07:52:54.1775721174
Is it valid to write irrational number written as an infinite sum of rational number?
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You can always write an irrational number as a sum of rationals. Then the last step in your argument depends on knowing that the function $f$ is continuous.
If you know that, then knowing that addition is preserved you can show $f(x) = cx$ for some constant $c$.
This is a well studied problem: see https://en.wikipedia.org/wiki/Cauchy%27s_functional_equation
I don't know whether assuming that the function preserves multiplication too suffices to get the continuity.