Definition of a solution of a differential equation?

Question

In one of the books, I found that, the definition stated as 'solution of the differential equation is

Given an open interval $I$ that consists to, a solution of the initial value problem $$ X' = f(X,t) ~~\text{ with }~~ X(t_o) = X_o \tag{1.1} $$ ['$t$' is the independent variable]

on $I$ is a differentiable function $x(t)$ defined on $I$, with $x(t_o) = x_o$ and $x'(t) = f(x,t)$ for all $t ϵ I$.'

My questions are,

1. What does open interval mean in this contest?
2. Why it can't be a closed interval?

Thanks in advance, Ram.

Answer 1

Since there doesn't seem to be any further discussion, I'm copying my comment above into an answer:

Just in general, non-open sets don't play nice with derivatives, since you wind up having to take them from just one side (or, in higher dimensions, from a restricted subset of all possible directions). In any case, a solution to an initial value problem can always have its domain extended to some maximal open interval where it will be a solution.