A Bayesian local smoother is proposed for the estimation of a smooth function. The regression function is modelled as the sum of a polynomial and a Gaussian random process as in classical splines. The problem is formulated as a general linear model and a hierarchical Bayesian approach is used to study it. An important feature of the proposed approach is that different degrees of smoothness can be assumed for the regression function in different regions of the domain but it is constrained to be continuous at the boundary points. The novelty of this approach is the implementation of this feature by modelling the continuity constraints in the prior covariance matrix. The main advantage here is that there is no need for the computationally intensive constrained minimization which is usually required. Instead the resulting estimator is expressed as a ratio of two single-dimensional integrals which can be evaluated (or approximated) easily. Additionally, the local fitting of the model provides additional structure to the covariance matrix of the data and hence leads to substantial computational efficiency. Credible bands are also developed for the regression function to reflect the uncertainty in estimation. Finally, a sensitivity study on the tuning parameter is performed and the methodology is illustrated with examples.