Continuity
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Uniform continuity theorem
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Let f be continuous on the closed interval [Equation]. Then for given there exists a [Equation] such that, if [Equation] are any two points of [Equation] such that [Equation] then [Equation]. Proof From the above lemma, divide the interval [Equation] into a finite number of segments where the leap of f is less than [Equation]. Let [Equation] be the length of the smallest of these segments. Given that [Equation] we have two possibilities: (i) [Equation] lie in the same interval, and (ii) [Equation] lie in adjacent intervals. In case (i) [Equation] and the theorem holds. In case (ii) let c denote the common end-point of the two subintervals, then [Equation] and the theorem holds.
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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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