removable discontinuity
Picture a smooth curve with a single pinprick hole, or one point lifted off and set down in the wrong place. A removable discontinuity is exactly that kind of harmless flaw: the function would be continuous if only you redefined it at that one point. The repair costs a single value.
Formally, f has a removable discontinuity at a if lim_{x->a} f(x) exists (a finite value L) but f is not continuous there — either because f(a) is undefined, or because f(a) is defined but does not equal L. Redefining f(a) := L produces a function continuous at a, ‘removing’ the discontinuity.
This is the mildest kind of failure of continuity, a first-kind discontinuity in which the two one-sided limits exist and agree. It is the situation behind every ‘hole’ you patch when simplifying expressions like (x^2 - 1)/(x - 1): the algebraic form is undefined at x = 1, but the limit there is 2, and filling in the value 2 heals it.
g(x) = (x^2 - 1)/(x - 1) is undefined at x = 1, but lim_{x->1} g(x) = 2. Setting g(1) = 2 makes it continuous — the discontinuity was removable.
Filling the hole with the limit value removes it.