Kneser's theorem on solution sets

Remarks

$S_c$ is nonempty by the Peano existence theorem. Compactness follows from the Arzelà–Ascoli theorem, since all solutions are bounded and $M$-Lipschitz. Connectedness is the heart of the proof: one approximates $f$ by functions for which solutions are unique. For these the value at time $c$ depends continuously on parameters, and one passes to the limit. Example: $x' = 2\sqrt{\abs{x}}$, $x(0) = 0$ has the solutions $x = 0$ and, for each $s \ge 0$, $x(t) = 0$ for $t \le s$ and $x(t) = (t - s)^2$ for $t \ge s$. So $S_c = [0, c^2]$, an interval, as the theorem predicts. In particular, the initial value problem has either exactly one solution on $[t_0, t_0 + \alpha]$, or some $S_c$ has more than one point and is then uncountable.

Going deeper

For a student

The theorem refines the Peano existence theorem: it describes the whole set of solutions, not just one. A stronger form says that the set of solutions itself, as a subset of the space of continuous functions, is compact and connected. These results are used in the theory of differential inclusions and in the study of non-unique flows.