推 bineapple :Thx 我會了~ 10/16 23:24
※ 引述《bineapple (パイナップル)》之銘言：
: f:R→R
: If there is at least one point x_0 in R where f is continuous on it,
: and f(x+y)=f(x)+f(y) for all x and y in R, show that f(x)=ax for some a.
: 我已經會證明當x_0不是0的情況了
: 想請問有高手能提示一下是0時的情況嗎??
: 謝謝
first of all, f is continue everywhere, since f(x+z)-f(x)=f(x_0+z)-f(x_0).
second, set a=f(1), then by linearality of f, then for all rational number
q, we have f(q)=aq.
Lastly, by continuity of f, we have for any r a real number, f(r)=ar.
Done.
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