Continuity comes from the via Old French, continuité, and Latin, continuus ("joining, connecting, following one after another") and the verb continere ("to be uninterrupted"), literally meaning "to hang together" (com- meaning together + tenere meaning to hold. 

According to the Oxford English Dictionary (OED), the noun continuity was first used in English in the mid-1500s.   Continue (verb/noun) Middle English period (1150–1500) / early 1500s.  Continuing (adjective) was first used in 1393 in the writings of John Gower.  Continuous (adjective) was furst used in 1642 in the writings of Thomas Fuller.  Continuously (adverb) was first used in 1678 in the writings of Ralph Cudworth.  

Just the facts:

CONTINUITY Definition & Meaning - Merriam-Webster

CONTINUITY Synonyms: 36 Similar and Opposite Words | Merriam-Webster Thesaurus

CONTINUOUS Definition & Meaning - Merriam-Webster

GAP Definition & Meaning - Merriam-Webster

As I was searching, I came across this one and just saying it made me chuckle, I'm betting you will too?

Lipschitz continuity - Wikipedia

Continuity in storytelling matters because it maintains audience immersion, preserves logical consistency, and builds trust between the author and the reader.  Without it, narratives feel disjointed and pull people out of the experience.  

Core Benefits of Continuity:  
Keeps Readers Engaged: Small errors or plot holes distract fans, making them focus on mistakes rather than the plot.  
Supports Character Growth: Consistent traits and histories help characters feel real and believable over time.  
Makes Worlds Believable: A stable timeline and matching physical rules make fictional settings feel like real places. 

Key Types of Consistency:  
Plot Continuity: Making sure past events, rules, and promises lead logically to current actions without direct contradictions.  
Visual and Spatial Continuity: Keeping physical objects, costumes, and settings in the correct place and style across different scenes.  
Temporal Continuity: Managing the passage of time accurately so events unfold in a sensible, chronological order.