Saturday, 8 March 2014

real analysis - Show that f(x+y)=f(x)+f(y) implies f continuous Leftrightarrow f measurable



Let f:RR, and for every x,yR 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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