A point at which a function is not defined or not regular but where the function can be redefined or extended so that it becomes regular (analytic or continuous as appropriate) in a neighborhood of the point.

Principle

A singularity is removable when the limiting behavior of the function near the point approaches a finite value (or the extension exists) allowing a unique redefinition that restores regularity without changing the function off the point.

Demonstration

The function f(z) = sin(z)/z on the complex plane has a removable singularity at z=0 because defining f(0)=1 makes f entire; the limit as z→0 exists and equals 1.

Misapplication

Mistaking a pole or essential singularity for removable leads to incorrect extensions; for example attempting to assign a finite value at a pole where the function diverges breaks analyticity and misleads residue computations.

Consequence

Identifying removable singularities allows extension of functions to larger domains, simplifies classification of singularities, and ensures correct application of theorems requiring analyticity or continuity on closed sets.

Reversal

The contrast classes are nonremovable singularities: poles (finite-order divergence) and essential singularities (wild non-polynomial behavior) where no redefinition yields regularity.

Boundary

Applies in contexts of analytic or continuous functions on topological or complex domains. Excludes removable-like redefinitions that violate required regularity (e.g., restoring continuity but not analyticity when analyticity is required).

Semantic Tension

Tension exists between 'removable' as a pointwise redefinition restoring continuity versus restoring stronger structures like differentiability or analyticity; what is removable depends on the sought regularity class.

Synthesis

A removable singularity is a defect in a function's domain that is only superficial: the function admits a well-defined extension at the point that restores the intended regularity, so the singularity can be 'filled in' without altering nearby behavior.