Continuous Images of Compact Sets

Real Analysis · Axiom Academy

LESSON Continuous Images of Compact Sets A continuous map carries a compact set to a compact set — and that one fact is exactly why a continuous function on [a,b] must attain a maximum and a minimum. 1. A Continuous Map Carries K to f(K) Take the compact domain — a closed segment with both endpoints included. Push every point of K through the continuous curve y=x^3-3x and read off the heights. The set of heights you land on is the image f(K) . Watch the closed segment K get carried onto another closed segment on the value axis. K — a compact (closed & bounded) domain f(K) — the image, also compact 2. Compact Means Closed and Bounded In , the Heine–Borel theorem says compact = closed + bounded. The image f(K)=[-2,2] passes both checks: a finite band traps it ( bounded ), and its two endpoints -2 and 2 are included ( closed ). Watch the band clamp in and the endpoints seal shut. The whole image fits inside a finite band — nothing runs off toward . The edge values -2 and 2 belong to f(K) — the endpoints are filled, not hollow. Closed and bounded in is exactly the Heine–Borel definition of compact. An open or unbounded set fails — its sup/inf can sit just out of reach. 3. Why It Works: Pulling Back an Open Cover

This is the written version of the interactive lesson above. See the full Real Analysis course.