Suppose that f:R→R 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 x→0. 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 a∈R 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 |x−a|<δ, we have |f(x)−f(a)|=|f(x−a)|<ϵ, therefore f is continuous at a.
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