"Qantas has the distinction of being the oldest airline in the English-speaking world, and has the second oldest surviving airline name in the world - KLM Royal Dutch Airlines being a year its senior."

http://users.chariot.net.au/~theburfs/qantaspage.html

Although KLM did not fly in the Netherlands whilst the country was occupied by Nazi Germany, operations continued in the Dutch East Indies until they were overrun by the Japanese. The airline maintained a network centered on the Dutch West Indies throughout the war, with flights to Trinidad, Barbados, Venezuela, Colombia, Surinam and Miami. This was known as KLM WIB (West Indisch Bedrijf). KLM also operated flights for NGAT (Netherlands Government Air Transport).

I'm inclined to believe this qualifies KLM as a "continuously operating airline".

KLM was established October 7, 1919; Qantas on November 16, 1920.

johan