TITLE: If $f$ is continuous and $f(x+y)=f(x)f(y)$, then $\lim\limits_{x \rightarrow 0} \frac{f(x)-f(0)}{x}$ exists
QUESTION [11 upvotes]: I'm solving the functional equation $f(x+y)=f(x)f(y)$ and I know that I have a continuous function $f:[0,\infty\rangle \to \langle 0,\infty\rangle$ s.t. $f(0)=1$. In one of the steps, I want to show that the limit $$\lim_{x \rightarrow 0} \frac{f(x)-1}{x}$$ exists and is finite?
I'm just looking for a hint.

REPLY [5 votes]: To get around the fact that you can't differentiate $f$ (yet), we'll integrate it! Define $F$ to be the following primitive of $f$
$$F(x) = \int_0^x f(t) \, dt$$
Integrating $f(x+y) = f(x)f(y)$, with regards to $x$, between $0$ and $1$ (actually you may chose any non zero number here), we get
$$F(y+1) - F(y) = f(y)F(1)$$
Since $f$ is positive, $F(1) \ne 0$, and we get
$$f(y) = \frac{F(y+1)-F(y)}{F(1)}$$
Now the right hand side is clearly $\mathcal{C}^1$, so $f$ is $\mathcal{C}^1$ too.