Continuity
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Uniform continuity
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Let [Equation goes here - download the original to see it.] and [Equation] on a closed interval [Equation]. The value [Equation] is called the leap for the function [Equation] in the interval. We now show that when a function is continuous on a closed interval then values of that function cannot “leap” or “jump” from one discrete value to another. This means that the function is uniformly continuous throughout the interval. The theorem that proves this is called the uniform continuity theorem. To prove it, we first require a lemma. Lemm Let f be continuous on the closed interval [Equation]. Then for any [Equation] then the interval can be subdivided into a finite number of segments in each of which the leap of f is less than [Equation]. Proof Suppose that the theorem is false. Bisect the interval [Equation]. Then in at least one half the theorem is false, and denoted it [Equation]. Iterate this process to obtain two sequences, one of increasing numbers [Equation] and another of decreasing numbers [Equation]. Each of these sequences converges on the same limit, which we shall denote by [Equation]. The function f is continuous at [Equation]. Hence, there exists an interval [Equation] such that the leap of f is less than [Equation]. But for sufficiently large n this interval [Equation] is included in [Equation], which is a contradiction.
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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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