sections of vector bundles transversal to a divisor

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?

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A section of $E$ cannot meet a point in $D$ transversally unless it avoids it. Is this what you have in mind ? –  Damian Rössler Apr 5 '13 at 7:52
I mean that for a point $x\in X$ and $s$ a section of $E$, either $x\in S$ and the $s(x)\notin D$, either $x\in X-S$ and then s(x) needs to meet transversally $D$. –  prochet Apr 6 '13 at 16:20