A point where a function is either undefined or defined with a value different from the finite limit of the function as the independent variable approaches that point, and where redefining the function value at that point to equal the limit makes the function continuous there.

Principle

A discontinuity is removable when the two-sided (or appropriate directional) limit exists and is finite; changing the pointwise value to that limit removes the discontinuity without altering the function elsewhere.

Demonstration

f(x) = (sin x)/x for x ≠ 0 and f(0) undefined has a removable discontinuity at 0 because lim_{x→0} (sin x)/x = 1; defining f(0)=1 makes f continuous at 0.

Misapplication

Calling a singularity removable when directional or two-sided limits do not exist (for example in the presence of a jump or an essential singularity) leads to incorrect redefinitions and loss of valid analytic information.

Consequence

A removable discontinuity can be fixed by a single local redefinition, restoring continuity and often enabling extension of analytic or differentiable structure if higher regularity conditions hold; however, derivative properties may not automatically extend.

Reversal

If the limit does not exist or is infinite, the discontinuity is not removable and must be classified as jump, infinite, or essential; those are the opposites of removability.

Boundary

Removability requires existence of the appropriate limit; it applies to real and complex single-variable contexts where limits are path-independent; in multi-variable or complex settings path-dependence or monodromy can prevent removability.

Semantic Tension

Distinguish removable discontinuity from a pointwise redefinition that preserves continuity versus more severe singularities like poles or essential singularities which cannot be corrected by a single redefinition; also contrast with jump discontinuities where one-sided limits differ.

Synthesis

A removable discontinuity is a local, finite mismatch between a function's assigned value and its limiting value that can be corrected by setting the point equal to the limit, thereby restoring continuity without altering the function elsewhere.