  We construct bootstrap confidence intervals for a monotone regression
function. It has been shown that the ordinary nonparametric bootstrap, based on
the nonparametric least squares estimator (LSE) $\hat f_n$ is inconsistent in
this situation. We show, however, that a consistent bootstrap can be based on
the smoothed $\hat f_n$, to be called the SLSE (Smoothed Least Squares
Estimator).
  The asymptotic pointwise distribution of the SLSE is derived. The confidence
intervals, based on the smoothed bootstrap, are compared to intervals based on
the (not necessarily monotone) Nadaraya Watson estimator and the effect of
Studentization is investigated. We also give a method for automatic bandwidth
choice, correcting work in Sen and Xu (2015). The procedure is illustrated
using a well known dataset related to climate change.
