I don't understand how it is possible to decide whether some function $y$ in a certain ODE involving dirac's delta and heaviside function is continuous without solving for y. Say for example:
$y'' +y' + y= (x-1)(\delta(x-3) + H(x-5) + H(x-1))$
I know it is possible to just solve for y using laplace transform, but I understood it is possible to know continuity for $y,y',y''$ just by looking at the expression. I will be really glad for help, I feel like I lack some basic understanding of these functions.
If $y$ satisfies the ODE then we need $y'$ and $y''$ to exist, so $y$ and $y'$ must be differentiable and hence continuous. I don't think that $y''$ will be continuous, though, because the sum of continuous functions is continuous and the right hand side of the equation is not continuous.