Sequential Characterization

Real Analysis · Axiom Academy

LESSON Sequential Characterization of Continuity A second way to read continuity — through convergent sequences — and the sharpest tool there is for proving a function is dis continuous. 1. The Sequential Characterization Let and let c be a limit point of D . The statement below says continuity is exactly the property that every input sequence marching to c has its output sequence march to f(c) . 2. Forward: Continuity Sequential Assume f is continuous at c and that . We thread the sequence through the ε–δ definition: continuity hands us a , convergence eventually drives every x_n inside that , and continuity then traps every image within of f(c) . 3. Reverse: Building a Witness Sequence We prove the converse by contrapositive: if f is not continuous at c , we construct a sequence whose images stay a fixed distance away. Shrinking the radius forces a fresh bad point at every step. 4. Catching a Discontinuity: at 0 To prove f is discontinuous at c you need just one sequence with . Take the sign function , with f(0) = 0 , and feed it x_n = 1/n . Let . For every n , , so . Hence , but f(0) = 0 . Since , f is discontinuous at 0 . One sequence settled it — no ε–δ argument required. 5. When the Images Won't Settle: at 0 Define for and f(0) = 0 . Here a single limit value never even forms: as the graph oscillates faster and faster between -1 and 1 . We expose it with two sequences that both run to 0 but whose images lock onto different heights.

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