Sunday, 10 September 2017

real analysis - Is a function f:mathbbRtomathbbR such that f(x+y)=f(x)+f(y) always continuous?

Is there a function f:RR such that f(x+y)=f(x)+f(y) which is not continuous? I have proved that if it's continuous in one point aR 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 xQ and f(x)=... if xRQ but I didn't find it.

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real analysis - How to find limhrightarrow0fracsin(ha)h

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