Solutions to ODEs

So apparently the (somewhat) formal definition for a solution to an ODE with independent and dependent variables x and y respectively is some y(x) such that F(x, y, y', y'' ... y'(n)) = 0 (where y'(n) is the nth derivative) for a range a < x < b (edit: actually surely this should allow for non-strict inequalities as well?). So does this mean if you solve a differential equation and obtain a function valid for more than one interval, for instance valid for a < x < b and b < x < c, you have two "solutions", one in each interval, or you have one solution that spans more than one interval? I guess what I'm asking is whether y(x) is a solution if and only if it is valid solely for some domain a < x < b, or if it is simply required to be valid for one interval but can in fact be valid for more than one.

So apparently the (somewhat) formal definition for a solution to an ODE with independent and dependent variables x and y respectively is some y(x) such that F(x, y, y', y'' ... y'(n)) = 0 (where y'(n) is the nth derivative) for a range a < x < b (edit: actually surely this should allow for non-strict inequalities as well?). So does this mean if you solve a differential equation and obtain a function valid for more than one interval, for instance valid for a < x < b and b < x < c, you have two "solutions", one in each interval, or you have one solution that spans more than one interval? I guess what I'm asking is whether y(x) is a solution if and only if it is valid solely for some domain a < x < b, or if it is simply required to be valid for one interval but can in fact be valid for more than one.