Continuity is uninterrupted connection, succession, or union — the property of something holding together unbroken across space or time.
The word makes a negative claim: it asserts the absence of a break. That is what distinguishes it from consistency, which asserts sameness. A practice can be perfectly consistent and still discontinuous — identical meetings resumed after a two-year gap — and perfectly continuous while changing throughout, as a river is. Continuity is a claim about the thread, not about the content strung on it.
Several fields have sharpened the word into a technical term. In mathematics, a function is continuous at a point when sufficiently small changes in the input produce arbitrarily small changes in the output; the nineteenth-century epsilon-delta formulation made that precise and remains the condition on which calculus depends. In film and television, continuity is the consistency of detail between shots recorded at different times, and a continuity error is its visible failure. In serial fiction, a continuity is the canon a body of stories shares, so that one franchise may carry several. In British broadcasting, continuity is the announcer's material between programmes. In business, continuity planning is the discipline of keeping operations unbroken through disruption.
English took the word through Middle French continuité from Latin continuitas, formed on continere, "to hold together" — con-, "together," plus tenere, "to hold." The etymology names the mechanism rather than the effect: continuity is not the absence of interruption by luck but the presence of something doing the holding.