Why can't I have ?
I'm in the very beginning of "Ordinary Differential Equations and Dynamical Systems" by Gerald Teschl and on page 9 he starts with differential equations of the form:
To solve these he quickly goes to:
Letting he points out that any solution must satisfy that . So, since F(x) is is strictly monotone in some neighborhood of , any solution will need to satisfy
Assuming, we can find a maximal interval where f is positive in this interval as well. So we can define:
So, . So far, so good. But, where I get confused is the following statement. Teschl states that:
In particular, is defined for all if and only if: