ZF cannot prove the local equivalence between "epsilon delta" continuity and "sequential" continuity for functions on the reals. A small fragment of (countable) AC does suffice for proving the equivalence.
In this light, most/many proofs in analysis would break down in ZF; a lot of work would be needed to sift through the details.
However, and in answer to your question, the aforementioned equivalence is provable in ZF for e.g. regulated functions. Thus, for the special case of regulated functions, no (extra) AC is needed anywhere.
A more general answer (based on recent research) is the following: it seems that countable AC can be avoided if we require that all functions under study are Baire 1 (or effectively Baire $n$).