The Ring of Continuation does not indicate what happens if you take it off. Does it simply end the effect, similar to casting a personal spell? Or if you cast a spell, took it off, and then put it back on, would the spell still be in effect?
While it is not specifically called out, the spell should revert to its normal duration as though you'd cast another personal spell (as is described in the item).
The normal behavior of magic items is that you lose their benefit immediately if they are removed.
It clearly would not be intended that you put the ring on in the morning, cast a spell, then swap to another ring for the rest of the day.