Help to solve $y'=y$, building exp function

Question

I come to ask for help building the exponential function as the solution to $y'=y$.

This question is different from :

Prove that $C\exp(x)$ is the only set of functions for which $f(x) = f'(x)$

Since I would like help to prove it using the following arguments :

1. show that the solution should verify : $f(a+b)=f(a)f(b)$
2. show that $f(x)$ for any $x$ in $\Bbb R$, will write $f(x)=c a^x$.
3. show that if the function value is $1$ at $0$, using a numerical tool we will be able to find the Euler constant value and not it e.

For the moment here are my ideas :

1. no idea – this is here that I need the more help
2. prove it for naturals, rationals then all real numbers using density arguments.
3. using Euler method,I can show that $a$ is the limit of $f(1) = \lim_{n\to\infty} (1+1/n)^n$ As you can see here, the computation will tend to $e$:

https://www.freecodecamp.org/news/eulers-method-explained-with-examples/

Many thanks, I'll appreciate your help

G

Answer 1

A hint for 1: consider the function $g$ defined as $$g(x)=\frac{f(a+x)}{f(x)}$$ and calculate $g'(x)$. What can you conclude?