Continuity and the Intrmediate Value Theorem
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Suppose that f is continuous on [a,b], f(z) < 0, and f (b) > 0. Set z = sup{x: f (t) < 0 for all t contained in [a, x]}. Prove that f (z) = 0. This is key to the proof of the intermediate value theorem. Incorporate the definition of least upper bound into your argument.
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Solution Summary
Continuity and the intermediate Value Theorem are investigated. The upper bound arguments are determined.
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Proof:
Let X={x:f(t)<0 for all t in [a,x]}. Then z=sup X.
From the definition, we know that for each x in X, x is also in [a,x] and thus f(x)<0.
Since z=sup X, we can find a sequence x_k in X, such that ...
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