Continuity Summary

Real Analysis · Axiom Academy

A function with no jumps, holes, or breaks — formalized by – , and the source of the existence theorems (IVT, EVT, Heine–Cantor) that power analysis. Continuity at a point: small changes in input force small changes in output — captured exactly by the – definition. Equivalent and often easier to check: the sequential view — . Continuity is preserved under sums, products, quotients (where the denominator is nonzero), and composition. On a closed bounded interval, continuity guarantees existence: the IVT (intermediate values / roots) and the EVT (a max and a min are attained). On a compact set, continuity upgrades to uniform continuity (Heine–Cantor); in metric spaces, continuous images of compact and connected sets stay compact and connected. f is continuous at a when every output tolerance can be met by some input tolerance . Intuitively: you can draw the graph through a without lifting your pen. Read it as: "outputs within once inputs are within ." Watch out for: here may depend on both and a . Core Concept Sequential Characterization Continuity is exactly the statement that f commutes with limits of sequences. This is frequently the cleanest tool — to disprove continuity, exhibit one sequence with . When to use: proving discontinuity, or continuity in metric spaces. Watch out for: the implication must hold for every sequence . Core Concept Algebra of Continuous Functions

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