Is there a function f:R→R such that f(x+y)=f(x)+f(y) which is not continuous? I have proved that if it's continuous in one point a∈R then it's continuous on all R, but I didn't find such a function which is not continuous everywhere. Therefore I tried to prove that all function of this form is continuous at x=0 but with no success. I think that if such a function exist it would be of the form f(x)=... if x∈Q and f(x)=... if x∈R∖Q but I didn't find it.
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