Let $V$ be a smooth vector field on $\mathbb{R}^n$. Assume that the maximal solution to the Cauchy problem $x'=V(x), x(0)=x_0$ exist only for $t\in [0,T)$, where $T$ is finite, denote this time by $T(x_0)$. Is $T$ continuous with respect to $x_0$ ? Is it $C^1$ ?
No, but you can say it is lower semicontiuous, even wrto the initial time (that is, the optimistic situation: perturbing a little the initial data the existence is ensured almost up to $T$ , and could even be much greater) . Precisely, given a Banach space $E$, an open set $\Omega\subset \mathbb{R}\times E$ and $f:\Omega\subset E\times \mathbb{R}\rightarrow E$ in the CauchyLipshitzPicard hypoteses, for any $(t_0,x_0)\in \Omega$ the Cauchy problem $$u(t_0)=x_0$$ for the ODE $$\dot u =f( t, u(t))$$ admits a maximal solution defined in an interval $\big(\tau_*(t_0,x_0), \tau^*(t_0,x_0)\big)\subset\mathbb{R}$, where $$\tau _ *:\Omega\to [\infty,0 ) $$ is upper semicontinuous and $$\tau ^ *:\Omega\to (0,+\infty]$$ is lower semicontinuous. This amount to saying that: the domain of the "general solution" $\xi:\Xi\subset \Omega\times\mathbb{R}\rightarrow E$ defined as $\xi(t_0,x_0,t):=u(t)$ with the solution $u(t)$ of the above Cauchy problem, that is the set $$\Xi:=\{ (s,x,t)\in \Omega\times\mathbb{R} \, :\, \tau _ * (s,x) < t < \tau ^ *(s,x) \}$$ (that is the zone between the graph of $\tau _ * $ and the graph of $\tau ^ *$), is an open set. 

