Exercise 1, § 1

Let $f$ be a continuous and differentiable map of $\mathbf{R}$ into a complete normed algebra $E$ over $\mathbf{R}$ having a unit element $e$; suppose that $f(0) = e$ and that, identically, $f'(x) = f(x)c$ where $c$ is an invertible element of $E$. Show that $f$ is a homomorphism of the additive group $\mathbf{R}$ into the multiplicative group of invertible elements of $E$. (Consider the largest open interval $I$ containing 0 such that $f(z)$ is invertible for all $z \in I$, and prove that for $x, y$ and $x + y$ in $I$ one has $f(x + y)(f(x)f(y))^{-1} = e$ keeping $x$ fixed and letting $y$ vary; then deduce that $I = \mathbf{R}$.)