Theorem: Every solution $y$ of the non-homogeneous equation $L[y]=f$ with $$L[y]=y''+py'+qy,$$ where $p,q$ and $f$ are continuous functions, is given by $$y=c_1y_1+c_2y_2+y_p$$ where the functions $y_1$ and $y_2$ are the fundamental solutions of the homogeneous equation, $L[y_1]=0,L[y_2]=0$, and $y_p$ is any solution of the non-homogeneous equation $L[y]=f$.
Why in the above theorem the functions $p,q$ and $f$ have to be continuous?
Thank you.