A family of kernel-based estimators parametrised by a bandwidth is used to estimate the location of the discontinuity in discretely observed data consisting of a discontinuous regression function with added noise. Asymptotic bias and variance calculations for the estimated location are given as a function of the bandwidth and the sample size. For functions which are smooth apart from a single discontinuity, bias is of order h2, and is not in general negligible compared with variance, which is of order h/n. The asymptotic mean square error rate MSE = O(n-4/3), obtained by balancing squared bias and variance, can usually be improved to O(n-2+δ) with δ > 0 arbitrarily small by choosing a smaller bandwidth h - O(n-1+δ). Special cases are treated and bias and variance are calculated explicitly. Only bias depends on the type of discontinuous function considered: For a step function with added noise and for any function which is antisymmetric about the discontinuity point, squared bias becomes negligible compared to variance, and MSE can therefore converge at the faster rate O(n-2+δ). For a smooth function with one added step and added white noise, bias is of order h3 showing that bias decreases more rapidly as the function becomes simpler.