We look into nonparametric regression with repeated measurements collected on
a fine grid. An asymptotic normality result is obtained in a function space.
This result can be used to build simultaneous confidence bands (SCB) for
various tasks in statistical exploration, estimation and inference. Two
applications are proposed: one is a SCB procedure for the regression function
and the other is a goodness-of-fit test for linear regression models. The first
one improves upon other available methods in terms of accuracy while the second
can detect local departures from a parametric shape, as opposed to the usual
goodness-of-fit tests which only track global departures. A numerical study is
also provided.