f(x+y)=f(x)+f(y) f(1)=c 证明f(x)=cx 满足f(x+y)=f(x)+f(y)的函数具体是什么函数，请...
f(x+y)=f(x)+f(y) f(1)=c \u8bc1\u660ef(x)=cx \u6709\u5956\u5178\u578b\u7684\u67ef\u897f\u65b9\u6cd5\u89e3\u51fd\u6570\u65b9\u7a0b.
\u4e00\u822c\u65b9\u6cd5\u662f\u5148\u5728\u6574\u6570\u96c6\u4e0a\u89e3\u8fd9\u4e2a\u51fd\u6570\u65b9\u7a0b,\u518d\u63a8\u5e7f\u81f3\u6709\u7406\u6570,\u6700\u540e\u7528\u6781\u9650,\u4e24\u8fb9\u5939\u4e4b\u7c7b\u7684\u65b9\u6cd5\u8bc1\u660e\u5728\u65e0\u7406\u6570\u96c6\u4e0a,\u8be5
\u89e3\u4e5f\u6210\u7acb.\u5373\u5b8c\u6210\u8bc1\u660e.
\u8be6\u7ec6\u8fc7\u7a0b:(\u53ea\u5bf9\u81ea\u53d8\u91cf\u5927\u4e8e\u96f6\u8bc1\u660e,\u5c0f\u4e8e\u96f6\u53ef\u4eff\u7167)
\u5bb9\u6613\u5f97\u5230f(nx)=nf(x)
\u4ee4x=1\u53ef\u4ee5\u5f97\u5230f(n)=nf(1)=cn.
\u6240\u4ee5\u5728\u6574\u6570\u96c6\u4e0a\u8be5\u51fd\u6570\u65b9\u7a0b\u7684\u89e3\u4e3af(x)=cx.
\u4ee4x=b/a,n=a(a,b\u4e3a\u4e92\u7d20\u6574\u6570).
\u53ef\u4ee5\u5f97\u5230f(b)=af(b/a)=cb
\u6240\u4ee5f(b/a)=c*b/a.\u5373\u5728\u6709\u7406\u6570\u96c6\u4e0a\u8be5\u89e3\u4e5f\u6210\u7acb.
\u6700\u540e\u662f\u65e0\u7406\u6570(\u6211\u9ed8\u8ba4\u8fd9\u4e2a\u51fd\u6570\u662f\u8fde\u7eed\u51fd\u6570):
\u5bf9\u65e0\u7406\u6570d,\u627e\u4e24\u4e2a\u548c\u5b83\u8db3\u591f\u63a5\u8fd1\u7684\u6709\u7406\u6570e,g
\u663e\u7136f(x)\u5355\u8c03\u9012\u589e.
e<d<g\u53ef\u4ee5\u6709f(e)<f(d)<f(g)
\u7531\u51fd\u6570\u7684\u8fde\u7eed\u6027\u53ea\u80fdf(d)=cd.\u8bc1\u6bd5.
\u8fd9\u6837\u7684\u51fd\u6570\u53ea\u6709\u6b63\u6bd4\u4f8b\u51fd\u6570\uff0c\u5177\u4f53\u5f62\u5f0f\u4e3a\uff1a
f(z) = kz k\u4e3a\u5e38\u6570
\u8bbe\uff1az=x f(x) = kx
z=y f(y) = ky
\u5f53 z=x+y \u65f6\uff0c
f(x+y) = k(x + y) = kx + ky = f(x) + f(y) \uff081\uff09
\u5176\u5b83\u51fd\u6570\u4e0d\u5177\u8fd9\u79cd\u6027\u8d28\u3002
\u8fd9\u79cd\u6b63\u6bd4\u4f8b\u51fd\u6570\u662f\u7279\u6b8a\u7684\u7ebf\u6027\u51fd\u6570(y=kx+b\uff0cb=0)\uff0c\u6ee1\u8db3\u53e0\u52a0\u539f\u7406\uff1a
\u5373\u5355\u72ec\u8f93\u5165\u7684\u7ed3\u679c\u7b49\u4e8e\u603b\u548c\u8f93\u5165\u7684\u7ed3\u679c\u3002\u8fd9\u5c31\u662f\uff081\uff09\u5f0f\u6240\u8868\u8fbe\u7684\u542b\u4e49\u3002
f'(x) ＝（f(x＋x0) -f(x))/x0 其中x0极限是0
f(x＋x0) 此式带你的条件f(x+y)=f(x)+f(y)
f'(x)＝f(x0）/x0 因为f(0)＝0 由罗必达法则
f'(x)＝f(x0）/x0 ＝f'(x0)
证明了它的导数是一个定值 再由f(1)=c f(0)＝0 证明f(x)=cx
f(b/a)+f(b/a)+...+f(b/a)(一共有a个)
=f(a*b/a)
=f(b)
前面得到f(nx)=nf(x)
另x=b/a,n=a就可得到
f(b)=af(b/a)
又有得到的f(x)=cx
则f(b)=cb
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