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intermediate value theorem∗
yark†
2013-03-21 12:35:24
If f is a real-valued continuous function on the interval [a, b], and x1 and
x2 are points with a ≤ x1 < x2 ≤ b such that f (x1 ) 6= f (x2 ), then for every y
strictly between f (x1 ) and f (x2 ) there is a c ∈ (x1 , x2 ) such that f (c) = y.
Bolzano’s theorem is a special case of this.
The theorem can be generalized as follows: If f is a real-valued continuous
function on a connected topological space X, and x1 , x2 ∈ X with f (x1 ) 6=
f (x2 ), then for every y between f (x1 ) and f (x2 ) there is a ξ ∈ X such that
f (ξ) = y. (However, this “generalization” is essentially trivial, and in order to
derive the intermediate value theorem from it one must first establish the less
trivial fact that [a, b] is connnected.) This result remains true if the codomain
is an arbitrary ordered set with its order topology; see the entry proof of generalized intermediate value theorem for a proof.
∗ hIntermediateValueTheoremi created: h2013-03-21i by: hyarki version: h30423i Privacy
setting: h1i hTheoremi h26A06i h70F25i h17B50i h81-00i
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