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Definition: Suppose f is discontinuous at a point c. Suppose the right and left hand limits of f(c), that is f(c+) and f(c-), respectively, equal each other, but do not equal f(c). Then c is called a removable discontinuity, since the discontinuity could be removed by redefining f at c to have the value f(c+)=f(c-).
Source: Mathematical Analysis, second edition by Tom M. Apostol