  The Nadaraya-Watson kernel estimator is among the most popular nonparameteric
regression technique thanks to its simplicity. Its asymptotic bias has been
studied by Rosenblatt in 1969 and has been reported in a number of related
literature. However, Rosenblatt's analysis is only valid for infinitesimal
bandwidth. In contrast, we propose in this paper an upper bound of the bias
which holds for finite bandwidths. Moreover, contrarily to the classic analysis
we allow for discontinuous first order derivative of the regression function,
we extend our bounds for multidimensional domains and we include the knowledge
of the bound of the regression function when it exists and if it is known, to
obtain a tighter bound. We believe that this work has potential applications in
those fields where some hard guarantees on the error are needed
