This paper is concerned with the approximation order of ENO-SR piecewise polynomial interpolatory techniques for piecewise smooth functions with isolated singularities. When the interpolatory pieces are of degree r, we propose a detection procedure based on differences of order r that allows us to prove that the global order of accuracy of the interpolatory reconstruction is improved by a factor of h relative to the linear methods, where h is the sampling rate, for isolated discontinuities in \(f^{(q)}\) and \(q \le r\) . In addition, we show that full accuracy is recovered for h below a critical threshold, that depends on the jump in \(f^{(q)}\) at the isolated singularities.