Suppose we have an EU or US patent with 3 claims. The examiner rejects one of the indipendent claims because of prior art. Ist it still possible to process with the patent application by giving up the indipendent claim that got rejected and only releasing the patent with the other 2 claims?
Absolutely, patent prosecution is a process. One may amend claim wording to narrow and avoid cited prior art, cancel some claims, or cancel all claims are submit a new set of claims. One may also argue that the cited reference is not valid prior art based on its date or not being in an analogous art, or not teaching all elements of the claim in question.