Continuous Images of Connected Sets

Real Analysis · Axiom Academy

LESSON Continuous Images of Connected Sets A continuous map can stretch and bend a set, but it can never tear it apart — and that one fact gives you the Intermediate Value Theorem for free. A set S is connected if you cannot split it into two nonempty pieces that are open in S , disjoint, and together make up all of S . Such a split is called a separation . The interval [a,b] is connected; the set is not — the gap is a separation. 2. Continuity Carries It Across Sweep a point across the connected domain and follow where f sends it. Because f is continuous, nearby inputs land at nearby outputs — so as the input traces its one unbroken piece, the output traces one unbroken piece too . No gap can ever appear in the image. Continuous image of a connected set is connected Suppose, for contradiction, the image did break apart. We carry that break backward through f and watch it tear the domain — which we assumed could not be torn. 4. The Intermediate Value Theorem Apply the theorem to the connected interval [a,b] in . The image f([a,b]) is connected — and the only connected subsets of are intervals. An interval has no gaps, so f attains every value between f(a) and f(b) . f continuous on [a,b] , and f(a) < y < f(b) some c gives f(c)=y You've seen that continuity cannot tear a set apart, why the proof works by pulling a separation backward, and how that single fact delivers the Intermediate Value Theorem. Scroll up to revisit any step.

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