The degradation of civil engineering structures may lead to a sudden stiffness reduction in a structure and such a sudden damage will cause a discontinuity in the dynamic responses. The detection on structural sudden damage has been actively carried out in this study. The signal singularity of the acceleration responses with sudden stiffness reduction is characterized by the coefficients of continuous wavelet transform with fine scales. A detection approach based on the CWT is proposed in terms of the decomposed detail coefficients of continuous wavelet transform to detect the damage time instant and location. The Lipschitz exponent is mathematically used to estimate the local properties of certain function and is applied to reflect the damage severity. Numerical simulation using a five-story shear building under different types of excitation is carried out to assess the validity of the proposed detection approach for the building at different damage levels. The sensitivity of the damage index to the intensity and frequency range of measurement noise is also investigated. The effects of both measurement noise intensity and frequency range on the damage detection are numerically investigated.