##  [Removable Singularity](/removable-singularity-0) 

 Definition

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

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

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

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

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

Reversal

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

 

 

 

 

 





## Boundary

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

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

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.