StellarA function is continuous at a point x=a if three conditions are all true. First, f(a) must exist. Second, must exist. Third, the limit must equal the function value. In symbols, continuity at a point means If any one of these conditions fails, then the function is not continuous at that point.
This definition is important because it separates several ideas students often mix together. A function might be defined at a point but still not be continuous there if the limit does not exist or if the limit and function value do not match. A function might also have a limit at a point but fail continuity because the function itself is not defined there. Continuity is therefore a combined condition involving both the limit and the actual function value.