Testing Continuity at x = 1

Calculus 1 · Axiom Academy

EXAMPLE Testing Continuity at x = 1 Check all three conditions for continuity at a single point of a piecewise function Determine whether the function below is continuous at x = 1 . If it is not, classify the discontinuity. Three Conditions for Continuity at x = a f(a) is defined — the function has a value at a . exists — the left- and right-hand limits agree. The limit equals the function value: . Nice work! You tested continuity at a point by checking all three conditions in order. Here's what carries forward: All three conditions must hold: even if two are satisfied, failing one means the function is discontinuous there. For piecewise functions: at a boundary point, compute the left- and right-hand limits separately — each side uses its own formula. The matching condition is the catch: the limit must equal the actual function value, not merely exist. Discontinuity types: removable (a hole), jump (left and right limits differ), and infinite (a vertical asymptote). Here both pieces meet at f(1) = 2 , so the graph has no break, jump, or hole — f is continuous at x = 1 .

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