Continuity
A function is continuous at a point only when all three things hold: the function is defined there, its limit exists there, and the limit equals the function's actual value there (no gaps alone is not the full definition). Consider a piecewise function where f(x) = x for every x except x = 3, and f(3) is separately defined as 10: the limit as x approaches 3 is 3 (following the f(x) = x rule from both sides), but f(3) itself is 10, so the limit does not match the function's actual value there, giving a removable discontinuity at x = 3.
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