Let X be acomplex infinite dimensional Banach space. We use σ_a(T) andσ_{ea}(T), respectively, to denote the approximate point spectrum and theessential approximate point spectrum of a bounded operator T onX. Also, \pi _a(T) denotes the set <$>{\rm{iso}σ_a(T)\backslash σ_{ea}(T)}<$>. An operator T onX obeys the a-Browder's theorem provided that<$>σ_{ea}(T) =σ_a(T\,)\backslash π_a(T)<$>. We investigateconnections between the Browder's theorems, the spectral mapping theorem and spectralcontinuity.