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
First step was to use induction to show that $\displaystyle f(n)=e^n$ 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 $\displaystyle f(-n) = 1/f(n) = 1/e^n = e^{-n}$.
Next, use induction on n to prove that $\displaystyle f(nx) = \bigl(f(x)\bigr)^n$, for any x and any positive integer n. It follows that if m and n are integers (with n positive) then $\displaystyle e^m = f(m) = f(n\tfrac mn) = \bigl(f(\tfrac mn)\bigr)^n$. Take n'th roots to see that $\displaystyle e^{m/n} = f(\tfrac mn)$. In other words, $\displaystyle f(r) = e^r$ for any rational number r.
Finally, use continuity to deduce that the result is true for all real numbers.