In the 1970s \textit{I. Král} [Séminaire de Théorie du Potentiel, Paris, 1972--1974, Lecture Notes in Math. 518, 95--106, Springer-Verlag, Berlin-Heidelberg, 1976 (1976; Zbl 0325.35012)] developed a general theory for removable singularities enabling him to characterize the removable singularities for solutions to semielliptic partial differential operators lying in BMO, VMO, Hölder classes, little Hölder classes and Campanato spaces. In this theory the functions are presumed to be in a function space on a domain \(\Omega\) and to be distributional solutions of the operator in \(\Omega\setminus E\), the question being when it can be deduced that the functions are distributional solutions of the operator in all of \(\Omega\). In this paper the author assumes the functions to be in the function space on \(\Omega\setminus E\) only. In this case one is lead to two different notions of removability: weak removability asking when the functions are solutions in all of \(\Omega\); and strong removability also requiring the extended solutions to lie in the function space on all of \(\Omega\). The main part of this paper is the development of the theory of weak and strong removable singularities for analytic functions such that \(\sup_{z\in \Omega} | f^{(n)}(z)| \,\text{dist} (z, \partial \Omega)^s<\infty\). Spaces of this type include spaces of holomorphic functions in Campanato classes, BMO and locally Lipschitz classes. The author obtains a number of examples with certain important properties. Similar examples have earlier been obtained for Hardy \(H^p\) classes and weighted Bergman spaces, mainly by the author. Because of the similarities in these cases, an axiomatic approach is used to obtain some results that hold in all cases with the same proofs.