The following are from pp.26~27, Section 1.5 of Partial Differential Equations by Michael E. Taylor.
Let $A(t)$ be a continuous, $n\times n$ matrix-valued function of $t\in I$. We consider the general linear, homogeneous ODE $$\frac{dy}{dt}=A(t)y,\quad y(0)=y_0.$$ It has a global solution. There is a matrix-valued function $S(s,t)$ such that the unique solution to the equation satisfies $$y(t)=S(t,s)y(s).$$
My question is: Why is $S(t,s)$ continuous in $t,s$? (In the next paragraph the author integrates $S(t,s)$.) I don't know where to start...