Непрерывность, собственно, и составляет предмет истории.

Such sentences are most difficult to translate. It is aphoristic and word-by-word translation will not work. Translator should be able to express an idea in the target language without reference to the original one.

There are many ways to translate "the very" depending on context, and none of them is universal.

Непрерывность - nominative.

Sentence is nominative since it answers the question what.- I don't understand this.

"свойство" will not work as 'stuff' here.

"история" is feminine and "of + smth." most often is translated as Genitive