  We consider nonparametric regression in the context of functional data, that
is, when a random sample of functions is observed on a fine grid. We obtain a
functional asymptotic normality result allowing to build simultaneous
confidence bands (SCB) for various estimation and inference tasks. Two
applications to a SCB procedure for the regression function and to a
goodness-of-fit test for curvilinear regression models are proposed. The first
one has improved accuracy upon the other available methods 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 of
the SCB procedures and an illustration with a speech data set are provided.
