As per MrAlien's answer for Why is it best if we light ghee lamp in the house while praying?, its based on Sanskrit saying,
असतोमा सद्गमय। तमसोमा ज्योतिर् गमया। मृत्योर्मामृतं गमय॥ ॐ शांति शांति
शांति - बृहदारण्यक उपनिषद् 1.3.28.
But for Akhanda Jyoti/Deepak,
Japa done in front of a akhand jyoti is 1000 times more effective.
Deities like Ganesha, hanuman, bhairav and Devi are pleased when
worshipped in front of a Jyoti. Akhand Deepak represents your
unshakable and unflickering faith in your Isht Devta.
Extinguishing of akhand deepak is considered inauspicious, so be
careful in maintaining it. This is an omen that there are upcoming
problems. If this happens, light the diya again and do the puja again
and increase the count of japa you regularly do to give you strength
to fight with the upcoming problem. ( Source:holydrops.com)
This is the same old saying I grown up listening.