continuation of solutions

A local existence theorem hands you a solution on a tiny interval, but that is rarely the end of the story. Continuation of solutions is the systematic act of extending that little local solution outward — gluing on more solution to the left and right — until you cannot extend it any further. It is how you pass from the modest local guarantee to the full lifespan of the trajectory, the maximal interval of existence.

The mechanism is a relay race. Suppose existence-and-uniqueness gives you a solution on (x0 - h, x0 + h). Take the endpoint value just before the right edge, treat it as a new initial condition, and reapply the local theorem there to extend a bit further; then repeat from the new frontier. Because the pieces overlap and uniqueness forces them to agree where they overlap, they patch together into one well-defined solution on a larger interval. You keep relaying until extension fails — and there is a clean dichotomy for why it stops: either the independent variable runs to the boundary of the region where f is defined, or the solution value leaves every bounded set (it blows up). If neither happens, you can always continue, so a solution that stays in a fixed bounded region must extend.

This is the bridge concept that makes 'maximal interval' precise and turns blow-up into a theorem rather than an accident. It also gives a useful working rule: to show a solution exists for all time, it is enough to show it cannot escape to infinity — an a priori bound on the solution forbids blow-up, and continuation then carries it forward forever. Gronwall's inequality is the standard tool for producing exactly such a bound.