There's a definition that says that if we have $\int_{a}^{c}f = \int_{a}^{b}f + \int_{b}^{c}f$ and at least one of $\int_{a}^{b}f, \int_{b}^{c}f$ diverges, then $\int_{a}^{c}f$ diverges. Why is this well defined? for example, how do we know that there is no function $f$ such that $\int_{0}^{\infty}f = \lim_{b \to \infty} \int_{0}^{b}f$ exists (finite), but if we write it as $\int_{0}^{1}f+\int_{1}^{\infty}f$ we get that $\int_{0}^{1}f$ diverges and therefore $\int_{0}^{\infty}f$ diverges?
EDIT: I realized the example I gave is actually impossible, but what about something like: $\int_{0}^{2}f$ converges, but both $\int_{0}^{1}f,\int_{1}^{2}f$ diverge?