Let f:R→R, and for every x,y∈R we have f(x+y)=f(x)+f(y).
Show that f measurable ⇔f continuous.
Answer
One implication is trivial. If a function is continuous, then it is measurable. The converse is more tricky.
You can find a very nice proof in the following document. Another proof can be found considering the function F(x)=∫x0f(t)dt, which is well defined since F is measurable.
Another approach is the following: prove that a discontinuous solution for the functional equation is not bounded on any open interval. It can be shown that for a discontinuous solution the image of any interval is dense in R, and therefore we have problems with the measurability.
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