Proving it is the only function that satisfies the properties

[Corvinus]

Hello everyone,

I've got an exercice in which I am asked to find the function that satisfies the following properties :
- f(0) = 1
- f(x + 1) = 2*f(x)
- f(m*x) = f(x)^m (with m being a member of the natural numbers)

So I found that the function f(x) = 2^x which I think is correct but now, I am asked to prove that this is the unique function by using the fact that it is continuous. Someone said me I would have to prove it with delta and epsilon. I think he was talkng about the same thing you can find in similar exercices when you have to prove that a function goes to a certain limit but I barely understand these concepts :/ . Can anyone help?

Thanks in advance,

Corvinus

 

[HallsofIvy]

"Using the fact that it is continuous"? Do you mean "show that the only continuous function satisfying these properties is 2^x"?

(f(x+h)- f(x))/h= f((1+h)x)- f(x))/h= (f(x)^{1+h}- f(x))/h= f(x)(f(x)^h- 1)/h. Now, taking the limit as h goes to 0, since f is continuous, we get f'(x)= af(x), where a= lim (f^h- 1)/h as h goes to 0, so that \(\displaystyle f(x)= Ce^{ax}\).

Now, we have \(\displaystyle f(0)= C= 1\) so \(\displaystyle f(x)= e^{ax}\). We also have \(\displaystyle f(x+ 1)= e^{a(x+1)}= e^{ax+ a}= e^ae^{ax}= 2f(x)= 2e^{ax}\) so that \(\displaystyle e^a= 2\).

Finally, we have \(\displaystyle f(x)= e^{ax}= (e^a)^x= 2^x\).

 

[Corvinus]

I don't understand how did you get from (f(x+h)- f(x))/h to f((1+h)x)- f(x))/h. Wouldn't it be f((1+h/x)x)- f(x))/h rather? I think I get the rest. I'm really impressed. I wouldn't have thought about it !

 

[HallsofIvy]

Oops! You are exactly correct. But, for each x, you can still write it as f'= af with \(\displaystyle a= \lim_{h\to 0}\frac{f^{h/x}- 1}{h}\)