Let $X$ a smooth projective curve over $\mathbb{C}$, $S$ a finite subscheme of $X$.

$E$ a vector bundle over $X$ with a divisor $D$. We look at the sections $A:=H^{0}(X,E)$ with $\deg E$ big enough.

Is it true, that the set $A^{0}_{S}$ consisting of sections that meet transversally $D$ and avoid $D$ at the points of $S$ is an open subset? And what should be the bound for $\deg E$ so that it holds?