Connected Sets and Continuity

Real Analysis · Axiom Academy

LESSON Connected Sets and Continuity A continuous function can stretch and bend a set but never tear it — so the image of one piece is one piece, and on the line that single fact IS the Intermediate Value Theorem. 1. One Piece In, One Piece Out Take a connected interval S on the domain line and a continuous f . As a marker slides across S , its image sweeps out the set f(S) on the value line. Watch what f(S) fills : because f never jumps, the marker's image traces a route that hits every value between its highest and lowest — so f(S) closes up into a single unbroken interval. No gaps appear. Worked example — the image is exactly an interval Let f(x)=x^ 3 -3x on the connected interval S=[-2,2] . Then f'(x)=3x^ 2 -3=0 at , and the values at the endpoints and critical points are f(-2)=-2 , f(1)=-2 , f(-1)=2 , f(2)=2 . The smallest is -2 , the largest is 2 , and every value in between is attained — so f(S)=[-2,2] , a single interval. Connected domain in, connected image out. 2. The Intermediate Value Theorem, For Free On the connected sets are exactly the intervals. So for continuous , the image f([a,b]) is connected — hence an interval — and an interval that contains both f(a) and f(b) must contain every value between them . Drop a horizontal target line at any height y between f(a) and f(b) : the curve cannot leap over it, so it must cross. That crossing is a point c with f(c)=y . a value between the endpoints is always hit Same curve, a target it must cross

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