Saturday, 15 February 2014

real analysis - Show a function for which f(x+y)=f(x)+f(y) is continuous at zero if and only if it is continuous on mathbbR




Suppose that f:RR satisfies f(x+y)=f(x)+f(y) for each real x,y.




Prove f is continuous at 0 if and only if f is continuous on R.




Proof: suppose f is continuous at zero. Then let R be an open interval containing zero. Then f is continuous at zero if and only if f(x)f(0) as x0. Then |f(x)f(0)|<ϵ.



Can anyone please help me? I don't really know how to continue. Thank you.


Answer



First let x=y=0, we have f(0)=2f(0) which means f(0)=0.
For any aR and a given ϵ>0, because f is continuous at 0, there exist δ>0 s.t. |x|<δ implies |f(x)f(0)|=|f(x)|<ϵ. Now consider the same δ, if |xa|<δ, we have |f(x)f(a)|=|f(xa)|<ϵ, therefore f is continuous at a.


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