As stated here, the answer is exactly the same as in the previous question: you can easily find a discontinuous function in any neighbourhood of a given continuous function.

However, this problem, unlike the previous one, makes sense (and is much more interesting) if you work entirely within C[a,b], the space of continuous functions on [a,b]. The reason is that this time the set A is not the whole space but a compact subset of it. So it makes sense to ask if A can contain an open subset of C[a,b]. I think that the answer is no, and that the reason is the

ArzeląAscoli theorem, which characterises the compact subsets of C[a,b]. I don't have time to think it through carefully just now, but I think that it is not possible for an open subset of C[a,b] to be equicontinuous. It follows from the AA theorem that such a set can never be compact in C[a,b].