Heine-Cantor Theorem

Real Analysis · Axiom Academy

LESSON The Heine-Cantor Theorem On a compact domain, continuity isn't just local — a continuous function is automatically uniformly continuous. 1. Continuity vs Uniform Continuity Both definitions ask the same thing — keep the output inside an -band — but they differ in where the may live. The animation fixes one and shows the largest that works at a flat point and at a steep point on the same curve. For each point x and each there is a — possibly depending on x — with . For each there is one — independent of x — so that for all x,y , . Continuity: the δ may move with x Uniform: one δ for every x at once 2. The Theorem: Compact ⟹ Uniform In this says: a continuous function on a closed, bounded interval [a,b] is automatically uniformly continuous — no extra assumption on f is required. The animation slides one -wide window all the way across [a,b] ; the output never leaves a single fixed -band, so the never has to collapse. On [0,3] , f(x)=x^2 satisfies . So for the choice works at every point at once — a positive, uniform . Both hypotheses — closed and bounded — earn their keep. Take f(x)=x^2 on the whole line (unbounded). It is continuous, but for a fixed output jump , the largest input gap that stays under it is about — and that shrinks toward 0 as . Watch the -interval narrow as it slides outward. Continuous · Uniformly continuous ✓ Continuous · Uniformly continuous ✗ 4. Drop Closedness: 1/x on (0,1)

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