Continuity Assumption

The continuity assumption is the identifying assumption of Regression Discontinuity: in the absence of treatment, the conditional expectation of the outcome would evolve smoothly through the Cutoff. No other variable can jump discontinuously at the threshold for any reason other than the treatment itself. Under continuity, any observed jump in $y$ at the cutoff is attributable to the treatment alone.

When to use

Continuity is plausible when units cannot perfectly sort across the cutoff. Defend it by (i) plotting pre-treatment covariates against the running variable — they should not jump at the cutoff (a Placebo Test); (ii) running the McCrary Density Test — the density of the running variable should be continuous at the cutoff (sorting would create a bunched-up density). If either test fails, the design is compromised.