Are the output functions of a continuous-time LTI system necessarily continuous (in the calculus sense) for any given input functions?

I had this question when I saw this claim in my textbook:

> 
> for the most part we will use the condition of initial rest for
> systems described by differential equations. In the example, since the
> input was 0 for t < 0, the condition of initial rest implied the
> initial condition y(0) = 0.
> 
> 
> 

$x(t) = 0$ if $t < 0$ -(initial rest)-> $y(t) = 0$ if $t < 0$

But to conclude that $y(0)$ itself is zero, $y$ needs to be continuous (at 0) given $x$.