I am solving a linear dynamical system $X'=A(t) X$, where $t$ is the independent variable and $A(t)$ is a square matrix. Some of the coefficients of $A(t)$ have a discontinuity at a certain value of $t$. I have a heuristic argument as to why the solutions of this system are continuous. Namely, it goes like this: if $X$ is discontinuous, then $X'$ will involve a $\delta$-function, which will not work if I insert it in my system. I am looking for a reference, which would give me theorems I could apply to show that the solutions are continuous.