ことなく

ことなく means "without doing": the action in front of it never happens while the main action goes on. It follows the dictionary form of a verb, passive included.

It stresses that the action did not happen at all, so いちど一度も and さいご最後まで often come with it. The verb is always positive; the "not" is in なく.

• × 諦めないことなく → 諦めることなく

ことなく, ずに and ないで

All three mean "without doing", and they differ in register. ないで is everyday speech, ずに fits speech and writing, and ことなく is formal and written.

ないで also covers "instead of" (バスに乗らないで歩いた) and requests (行かないで). ことなく does neither.

ことなく and ことなしに

ことなしに also means "without doing", but it most often comes with a negative second half: without X, Y cannot happen. ことなく mostly reports that X did not happen.