Though it was very late, they continued their discussion.
I interpret continue as “keep discussing without stopping”.
Can I use continue with interchangeably?
To continue something is to keep doing it or to resume doing it. The activity is not terminated but prolonged.
They continued their discussion long into the night.
They continued their hike around the mountain lake after stopping for a brief rest.
To continue with something is to not change or alter it in response to conditions, but to proceed with it, as before. The complement of with in this pattern will be some sort of undertaking which is instrumental in achieving a result or goal of some kind.
The sky began to get gray, but the weather app said only light showers were expected, so they continued with their plans to have a picnic later that afternoon.
Notwithstanding the opposition they encountered from the National Rifle Association, the students continued with their efforts to ban the sale of military assault weapons.