A bounded linear operator \(T\) on an infinite-dimensional separable complex Hilbert space \(H\) is said to satisfy a-Weyl's theorem (a for ``approximate'') if the set of points \(\lambda\) in its approximate point spectrume such that \(T-\lambda\) has closed range, finite-dimensional null space, and non-positive index equals the set of those isolated points in the approximate point spectrum which are eigenvalues of finite multiplicity. Each of the following conditions is sufficient to imply a-Weyl's theorem for each \(f(T)\), where \(f\) is an arbitrary holomorphic function in an open neighborhood of \(\sigma(T): T^*\) is \(p\)-hyponormal \(((TT^*)^p\geq (T^*T)^p)\) or log-hyponormal (i.e. \(T^*\) is invertible and \(\log(TT^*)\geq \log(T^*T))\) or \(M\)-hyponormal (i.e. \(M\|(T^*- z)x\|\geq\|(T- z)x\|\) for some \(M> 0\) and all \(z\in\mathbb{C}\), \(x\in H\)) or \(T\) is reduced by each of its eigenvalues and is reduction approximate-isoloid (i.e. for the restriction of \(T\) to any reducing subspace each isolated point in the approximate point spectrum is an eigenvalue). Moreover continuity results of the approximate point spectrum are shown.