You left out some important information (like how Month is stored) and what database you're using, but here's a logical approach that you can start with:
CREATE VIEW CustomerBilledInMonth (CustName, Month, AmountBilled, ContinuousFlag) AS
SELECT CustName, Month, SUM(AmountBilled), 'Noncontinuous'
FROM BillingTable BT1
WHERE NOT EXISTS
(SELECT * FROM BillingTable BT2 WHERE BT2.CustName = BT1.CustName AND BT2.Month = BT1.Month - 1)
GROUP BY CustName, Month
UNION
SELECT CustName, Month, SUM(AmountBilled), 'Continuous'
FROM BillingTable BT1
WHERE EXISTS
(SELECT * FROM BillingTable BT2 WHERE BT2.CustName = BT1.CustName AND BT2.Month = BT1.Month - 1)
GROUP BY CustName, Month
Assuming that Month here is a consecutive integer field incremented by one from the first possible month in the system, this gives you with each customer's billing for each month summed up, and an additional flag containing 'Continuous' for those months that followed a month in which the customer was also billed and 'Noncontinuous' for those months that followed a month in which the customer was not billed.
Then:
SELECT CustName, LISTOF(Month), SUM(AmountBilled), MAX(ContinuousFlag)
FROM CustomerBilledInMonth GROUP BY CustName
will give you more or less what you want (where LISTOF is some kind of COALESCE type function dependent on the exact database you're using).