Removability

Removability is a concept used in several areas of mathematics to describe when a problematic region, such as a point or a small set, can be ignored without altering a desired property of a function or object. In many contexts, a region is removable if every function that is defined and satisfies a property on the surrounding domain can be extended across the region while preserving that property.

In complex analysis, removability is a central idea for extending analytic structure. A point a is a removable singularity of a function f that is holomorphic on a punctured neighborhood of a if f is bounded near a (equivalently, if the limit of f as z approaches a exists). In this case, f extends to a holomorphic function at a. More generally, a subset E of the complex plane is called removable for a class of holomorphic functions if every function holomorphic on a domain minus E and bounded near E extends holomorphically over E. Characterizations of removability for larger sets involve advanced concepts such as analytic capacity; a compact set with zero analytic capacity is removable for bounded holomorphic functions, a subject linked to the Painlevé problem.

In real analysis, removability often refers to removable discontinuities. A point where a function is undefined or discontinuous is removable if the limit exists as the variable approaches that point, allowing the function to be redefined so that it becomes continuous at that point.

Beyond complex and real analysis, removability also appears in topology and potential theory, where the question is whether a set is removable for broader classes of maps or for harmonic, Lipschitz, or other structured functions. Across contexts, removability addresses when local properties extend globally across a region.