I am studying differential equations using MIT's publicly available materials. One of the exercises runs as follows:
Let $I = (a,b)$ be an open interval containing $0$, and consider the ODE \begin{align} y^{(n)} + a_{n-1}y^{n-1} + \cdots + a_ny = f(t) \end{align} where has a finite number of simple discontinuities on $I$ and $|f(t)| \leq Ae^{Bt}$ for all $t \in I$ where it is defined for some constants $A,B \in \mathbb{R}^+$. Moreover, let $f$ be continuous at $0$. Establish the existence and uniqueness theorem for the IVP $y(0) = y_0, y'(0) = y_1, \ldots, y_{(n-1)}(0) = y_{n-1}$ in the class of so-called generalized solutions of the ODE; i.e. solutions $\phi$ where $\phi, \phi', \ldots, \phi^{(n-1)}$ are all continuous on $I$ and $\phi(t)$ solves the ODE whereever $f$ is continuous.
Here's what I have so far: We may assume without loss of generality that $f$ is discontinuous at only one point $c$, and that $c > 0$. Let $I_0 = (a,c)$, $I_1,(c, b)$. Since $Ae^{Bx}$ is Lipschitz on $(a,c)$, $f$ must be as well, and so there exists a unique solution $\phi$ to our IVP satisfying our continuity conditions on $I_0$. The difficulty, for me, consists in 'extending' $\phi$ to $I_1$; it seems to me that we don't have enough information about the behavior of $y$ on $I_1$ --say, initial conditions-- to know that $\phi$ is going to be continuous (and have the requisite continuous deriviatives) at $c$.
Any help would be appreciated.