I have the following statement of the inverse function theorem from Munkres' Analysis on Manifolds:
(IFT) Let $A$ be open in $\mathbb{R}^n$; Let $f:A\to\mathbb{R}^n $be of class $C^r$. If $Df(x)$ is non-singular at the point $a$ of $A$, then there is a neighborhood $U$ of the point $a$ such that $f$ carries $U$ in a one-to-one fashion onto an open set $V$ of $\mathbb{R}^n$ and the inverse function is of class $C^r$.
I'm trying to understand the following example for the solution of an ODE $$ x'(t) = f(x), x(t_0) = x_0 $$ where $f:(a_1, a_2)\to\mathbb{R}$ is continuous and isn't zero at any point of $(a_1, a_2)$:
If $\varphi$ is a solution, then $\varphi'(t) = f(\varphi(t))$ and $\varphi(t_0) = x_0$. Then $\frac{\varphi'(t)}{f(\varphi(t))} = 1$. If $F:(a_1, a_2)\to\mathbb{R}$ is given by $F(x) = \int_{x_0}^x\frac{1}{f(s)}ds$ then $F'(x) = \frac{1}{f(x)}\neq 0$ in $(a_1, a_2)$ therefore $F$ is invertible and takes $(a_1, a_2)$ to an interval $(b_1, b_2)$ where $F^{-1}$ is defined. Therefore $1 = \frac{\varphi'(t)}{f(\varphi(t))} = F'(\varphi(t))\varphi'(t) = (F\circ\varphi)'(t)$ and integrating both sides we have $F(\varphi(t)) - F(\varphi(t_0)) = t-t_0$ but $F(\varphi(t_0)) = 0$ therefore $F(\varphi(t)) = t-t_0$ then the solution is given by $\varphi(t) = F^{-1}(t-t_0)$ and we can clearly see that it will be the only solution.
In the highlighted part they seem to have used the IFT but in the statement I read from Munkres it doesn't guarantee that $F$ would have a inverse in all of $(a_1, a_2)$. It would only be locally one-to-one for all $x\in(a_1, a_2)$ such that $F'(x)\neq 0$. So we can't say in general there will be only one inverse $F^{-1}:(b_1, b_2)\to(a_1, a_2)$.
Can someone help me clarify this? Thanks in advance.
The assertion of the general inverse function theorem holds in $\mathbb R^n$. For $n=1$ there is a stronger assertion: If $f'(x)\ne0$ for all $x$ on some interval and $f'$ is continuous, then necessarily $f'$ does not change sign on this interval, that is, $f$ is either strictly increasing or strictly decreasing on that interval and thus (globally) one-to-one on that interval.