Re: [微積] f(x+y) = f(x)f(y) implies f(x) = a^x

[jacky7987 (憶)]

※ 引述《GSXSP (Gloria)》之銘言：
: f(x) differentiable on (-∞,∞)
: with f(x+y) = f(x)f(y)
: prove that f(x) = a^x for some a
: (Hint: ln f(x) must have constant derivitive)
: 我做                f'(x)       f'(y)
: grad ln f(x+y)  =  (------ ,   ------ )
:                      f(x)        f(y)
:                  f'(x)
: 但是還是看不出來 -----  是constant
:                  f(x)
: 請問這題要怎麼做呢?

提供另外一個方法試試看好了
                2
f(0)=f(0+0)=f(0)
=>f(0)=0 or 1

Case 1:
If f(0)=0 them f(x)≡0 for all x in R.....done
Case 2:
if f(0)=1
for n \in N           n
f(n)=f(1+1+...+1)=f(1)
       ￣￣￣￣n times

1=f(0)=f(1+(-1))=f(1)*f(-1)

             -1
=> f(-1)=f(1)

Hence for n be negtive integer
                                        -n    -1     -n      n
f(n)=f((-1)*(-n))=f(-1)*f(-n)=f(-1)*f(1)  =f(1) * f(1)  =f(1)
                  1     1         1        1  m
Moreover, f(1)=f(--- + --- + ...+---) = f(---)
                  m     m         m        m
                 ￣￣￣￣￣￣￣￣m times


                   1
                 (---)
         1         m
hence f(---)=f(1)
         m
                        1        n
                      (---)     ---
            n      1    n        m
finally, f(---)=f(---)      =f(1)
            m      m

到此我們做完全部的有裡數的部份
therefore, for r be a irrational number there exists a seq {x_n} ,where x_n
are all rational number such that x_n->r

and f is diff => f is conti

Hence
                                    x_n      r
f(r)=lim f(x)= lim  f(x_n)= lim f(1)    =f(1)
     x->r     n->inf       n->inf

let f(1)=a, we are done
＝＝＝
這作法是模仿線性的XDD
f(x+y)=f(x)+f(y)

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◆ From: 114.37.124.220
※ 編輯: jacky7987       來自: 114.37.124.220       (02/12 13:40)
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