This thesis addresses the issue of detecting discontinuities in real-time observational data from plant and machinery. This is highly relevant, since systems whose dynamics are well modelled by differential equations should exhibit continuity in the real-time signals and their derivatives. This work builds upon previous research into the detection of $C^n$ discontinuities and extends it to a more general case. Considering a set of $n$ derivative orders, the new approach permits defining which of these $n$ orders are to be inspected for discontinuities. All the derivations for the method, based on matrix algebraic formulations, are provided. Furthermore, a numerical solution for the method is implemented. Testing has been performed with a wide set of data sets derived from strongly differing areas of application. Performance estimates are computed using two different metrics and are compared with the results from other discontinuity and change detection methods. This new approach is the most generic of all the methods considered, since it does not require application specific adaption. It is based on a formal mathematical definition of a discontinuity, which is a generalization of a $C^n$ discontinuity. The comparative results show that the algorithm, on average, outperforms the other methods. There are, however, specific cases where the application specific methods perform better. Previous literature and test datasets only consider $C^0$ and $C^1$ type discontinuities. Whereas, the new approach, not only performs well for these type of discontinuities, but also functions for higher order derivative discontinuities.