Continuity
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The intermediate value theorem
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Let [Equation] be continuous in the closed interval [Equation] and that [Equation]. Then [Equation] takes every value that lies between [Equation]. Proof Suppose [Equation] Let [Equation] be a number such that [Equation] We have to show that there is a number [Equation] such that [Equation]. Let S be the set of numbers x in [Equation] such that [Equation]. This set is not empty since [Equation]. Therefore, it has a supremum. (This follows from the following theorem that employs the Dedekind cut: if S is a non-empty set of numbers that is bounded above, then S has a supremum.) Let [Equation]. 1. Since f is continuous at a there is an interval [Equation] on which [Equation]. Hence [Equation]. Likewise, there is an interval [Equation] on which [Equation], and hence [Equation]. Hence [Equation] 2. Since [Equation] for every [Equation] there is an element [Equation] such that [Equation]. For this [Equation], [Equation]. Since f is continuous at [Equation] we have [Equation] 3.By a similar argument [Equation] 4. Then [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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