The Intermediate Value Theorem

Real Analysis · Axiom Academy

LESSON The Intermediate Value Theorem If a continuous function starts below a value and ends above it, it must hit that value somewhere in between — there is no way to skip past it. 1. The Theorem, and the Line It Must Cross Let f be continuous on the closed interval [a,b] . Pick any target value y lying between f(a) and f(b) . The claim is an existence statement: some input c produces it. f continuous, y between the endpoint heights then a point c with f(c)=y is guaranteed to exist Picture a horizontal line at height y . The graph starts below it at a and ends above it at b . A continuous curve cannot leap from one side of a line to the other — so it must cross the line, and the crossing is a c with f(c)=y . 2. Continuity Is What Does the Work Drop the continuity hypothesis and the conclusion collapses. A function with a jump can step right over a value without ever taking it — the very gap that continuity forbids. Take a function equal to 1 up to the jump and 3 after it. Both endpoint heights exist, and y=2 sits squarely between them — yet nothing on the graph ever has height 2 . The jump skips it. No breaks. To get from below the line to above it, the graph is forced through the line — every intermediate value is attained. The graph teleports across the gap. Values inside the gap, like y=2 here, are simply never reached.

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