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
Now think about f(-n). The defining relation says that f(-n)f(n) = f(-n+n) = f(0) = 1, so .
Next, use induction on n to prove that , for any x and any positive integer n. It follows that if m and n are integers (with n positive) then . Take n'th roots to see that . In other words, for any rational number r.
Finally, use continuity to deduce that the result is true for all real numbers.