Unformatted text preview: ) = f ( y ), there is nothing to prove as that value is attained twice. Thus, assume f ( x ) > f ( y ). If f (0) < f ( x ), then ﬁx k ∈ R so that f (0) < k < f ( x ) and f ( y ) < k < f ( x ). Then by the intermediate value theorem, there exist points p between 0 and x and q between x and y where f ( p ) = f ( q ) = k . Thus f takes on a value twice. Similarly if f (0) ≥ f ( x ), then f (1) > f (0) ≥ f ( x ) > f ( y ) Thus we can choose k so that f ( y ) < k < f ( x ) and f ( y ) < k < f (1) and argue similarly. 1...
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 Spring '10
 MetCalfe
 Calculus, Intermediate Value Theorem, Continuous function, J. Metcalfe

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