This paper develops a local asymptotic theory for certain functionals of moving averages of Ito semi-martingales plus noise when the semimartingales are nearly continuous. The model provides a more complete interface between the continuity and jump asymptotics developed in (Jacod, Podolskij, and

Vetter 2009). Simulation suggests that our theory provides an improvement over the existing theory in a

practically relevant setting. The theory has applications to the estimation of functionals of jumps and to

the analysis of the local asymptotic power of tests for jumps with noisy high frequency data.