Example 3

For the scalar equation $x' = 1 + x^2$ one has $I = H = \mathbf{R}$: the integral that vanishes at $t = 0$ is $\tan t$, and the largest interval containing 0 where this function is continuous is $J = ] - \pi/2, +\pi/2[$; and $|\tan t|$ tends to $+\infty$ at the endpoints of $J$ (cf. IV, p. 173, cor. 1).