In this paper, we present a constructive method for the stability analysis of the bounded extremum seeking (ES) with discontinuous dithers and measurement bias, by using a time-delay approach to averaging. Discontinuous dithers used in ES are important for digital implementation. For conceptional clearness, we consider bounded ES of static maps in the case of single-input. By transforming the bounded ES dynamics into a time-delay system where the delay length is equal to the period of dither signal, we derive the efficient practical stability conditions for the resulting time-delay system in the form of linear matrix inequalities (LMIs). The stability of the transformed time-delay system guarantees the stability of the original ES plant. Under the assumption of the uncertain extremum point with the uncertain Hessian, the time-delay approach gives a quantitative calculation on the lower bound of the dither frequency and the upper bound on the resulting ultimate bound of seeking error. Fundamentally different from the classical averaging and Lie bracket methods for extremum seeking which are of approximate nature, the time-delay method proposed in the paper does not use any approximation via traditional averaging or Lie-bracket-based averaging.