Continuity
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Existence theorem for an inverse function
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Let f be continuous and strictly increasing on a closed interval [Equation]. Let [Equation]. Then there exists a function g that is continuous and strictly increasing on [Equation] such that [Equation]. That is, [Equation] the inverse of [Equation]. Proof Let k be any number such that [Equation]. By the intermediate value theorem, there exists an h such that [Equation] Since f is strictly increasing this value of k must be unique. Then the inverse function g is defined by [Equation]. To show that g is strictly increasing, suppose [Equation]. Suppose [Equation] then, since f is increasing, [Equation], hence [Equation], which contradicts [Equation]. Hence, [Equation], so g is strictly increasing. To show that g is continuous, let [Equation] and [Equation] Then, since f is increasing [Equation] and [Equation] implies [Equation] This applies to all values of [Equation], so g is continuous at [Equation].
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Contents of
Continuity
1 Continuity
2 Properties of continuous functions
3 The intermediate value theorem
4 Bounds of continuous functions
5 A continuous function in a closed interval attains its bounds
6 Uniform continuity
7 Uniform continuity theorem
8 Inverse functions
9 Existence theorem for an inverse function
10 Maxima and minima
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