A general one-dimensional discrete monatomic model is investigated by using the multiple-method. It is proven that the discrete bright breathers (DBBs) and discrete dark breathers (DDBs) exist in this model at the anti-continuous limit, and then the concrete models of the DBBs and DDBs are also presented by the multiple-scale approach (MSA) and the quasi-discreteness approach (QDA). When the results are applied to some particular models, the same conclusions as these presented in corresponding references are achieved. In addition, we use the method of the linearization analysis to investigate this system without the high order terms of ?. It is found that the DBBs and DDBs are linearly stable only when coupling parameter ÷ is small, of which the limited value is obtained by using an analytical method.