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