Given, simple model, $$\begin{eqnarray} \frac{\mathrm{d}S}{\mathrm{d}t} &=& -\beta SI \\ \nonumber \frac{\mathrm{d}I}{\mathrm{d}t} &=& \beta SI - \gamma I \\ \nonumber \frac{\mathrm{d}R}{\mathrm{d}t} &=& \gamma I \end{eqnarray}$$
Is variable $S$ continuous with respect to $\beta$, if one vary $\beta$?
My thoughts : if I vary $\beta$ very small, then rate of change become also small, but i doubt is it possible only for after small amount of time or for after large amount of time also.
Is there suitable way to describe it?