The majority of structural health monitoring studies reported in the technical literature focus on identifying damage sensitive features that can be extracted from dynamic response data. However, many of these studies assume the structure can be modeled as a linear system before and after damage and use parameters of these models as the damage sensitive features. The study summarized in this paper proposes a damage sensitive feature that takes advantage of the nonlinearities associated with discontinuities introduced into the dynamic response data as a result of certain types of damage. Specifically, the Holder exponent, a measure of the degree to which a signal is differentiable, is the feature that is used to detect the presence of damage and when that damage occurred. A procedure for capturing the time varying nature of the Holder exponent based on wavelet transforms is demonstrated through applications to non-stationary random signals with underlying discontinuities and then to a harmonically excited mechanical system that contains a loose part. Also, a classification procedure is developed to quantify when changes in the Holder exponent are significant. The results presented herein show the Holder exponent to be an effective feature for identifying damage that introduces discontinuities into the measured dynamic response data.