Can a lender continue foreclosure action if you pay the arrearages and the payments are current?

If foreclosure has been started, it doesn't really matter if you pay the loan up to current. Your original contract probably stated that once you are in default the entire amount can be called due, and the property can be sold for the amount owed (foreclosure).

As soon as you are in default, the bank can foreclose. If you pay the past due amount, they MAY decide to let the loan continue. It's the bank's decision. In general, once you have been in default you are considered to be a higher risk. The bank may let you re-finance the loan, but they also may want to charge a higher interest rate because you are no longer a low risk borrower.