Continuity of second derivate respect a first derivate in a differential equation. > Let $E$ a open subset of $\mathbb R^n$ containing $X_o$ and suppose $f \in C^1(E)$. Let $x(t)$ the solution to the PVI: $$ x'=f(x), x(0)=x_0 $$ in a interval $I$. Prove that the second derivative $x''(t)$ is continuous in I. I'm trying to prove this, but all I do is digress with the definition of continuity, can anyone help me, maybe a hint? thanks.

Just use that in $$ x'(t)=f(t,x(t)) $$ the right side, and thus also the left, is once continuously differentiable in $t$ as composition of $C^1$ functions.