# Why can't a union of two intervals be the maximum existence interval of a solution?

Assume the IVP $$x'=x^2, \quad x(0)=1$$ By separating the variables and using the initial condition $$x(t)=(1-t)^{-1}$$ Then, my book states that since $$x_0=1>0$$ then the solution can only be extended to the left with maximum existence interval $$\, I=(-\infty,1)$$.

My question is, since the solution $$x(t)$$ we found is defined for every $$t\neq1$$, why isn't the maximum interval of its existence $$I^*=(-\infty,1)\cup(1,+\infty)$$? Why does it depend on the sign of $$x_0$$?

With the given initial condition, $$(-\infty, 1)$$ is as far as you go without losing continuity.