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Math Help - Exonential function

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    Exonential function

    How to show that if f is continuous ,f(0)=1,f(1)=e,f(x+y)=f(x)f(y) then f(x)=e^x
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    Quote Originally Posted by drpawel View Post
    How to show that if f is continuous ,f(0)=1,f(1)=e,f(x+y)=f(x)f(y) then f(x)=e^x
    Try using Power series
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    Quote Originally Posted by drpawel View Post
    How to show that if f is continuous ,f(0)=1,f(1)=e,f(x+y)=f(x)f(y) then f(x)=e^x
    Do it in stages. First, f(n) = e^n for a positive integer (by induction starting with n=1). Then you'll have to think about why it follows for a negative integer, and then for any rational number. Finally, use continuity to get it for any real number.
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    I was trying to prove it but it seems that I just cannot handle this problem.Can you help me more.
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  5. #5
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    Quote Originally Posted by drpawel View Post
    I was trying to prove it but it seems that I just cannot handle this problem.Can you help me more.
    First step was to use induction to show that f(n)=e^n if n is a positive integer. I hope you can manage that (you're certainly not getting any more hints on that part).

    Now think about f(-n). The defining relation says that f(-n)f(n) = f(-n+n) = f(0) = 1, so f(-n) = 1/f(n) = 1/e^n = e^{-n}.

    Next, use induction on n to prove that f(nx) = \bigl(f(x)\bigr)^n, for any x and any positive integer n. It follows that if m and n are integers (with n positive) then e^m = f(m) = f(n\tfrac mn) = \bigl(f(\tfrac mn)\bigr)^n. Take n'th roots to see that e^{m/n} = f(\tfrac mn). In other words, f(r) = e^r for any rational number r.

    Finally, use continuity to deduce that the result is true for all real numbers.
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