Answer:
object of the preposition
Explanation:
The word mistake is within the prepositional phrase despite the mistake. In that matter, prepositional phrases are a number of words which begin with a preposition and contain an object and modifiers. Besides, they can function as adjetival phrases when they modify nouns, and as adverbial phrases when they modify verbs. In the sentence given, the prepositional phrase despite the mistake works as an adverbial phrase, since it modifies the verb continued.