Continuity
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A continuous function in a closed interval attains its bounds
Equations are omitted for technical reasons - download the original pdf
If f is continuous for [Equation goes here - download the original to see it.] and [Equation] then there exists an [Equation] such that [Equation]. (A similar statement could be written for the infimum.) Proof by contradiction Suppose there is not an x for which [Equation] and [Equation]. The for all [Equation], we have [Equation]. Since [Equation] the function [Equation] is continuous for [Equation]. Therefore it is bounded on a closed interval and by theorem (1) above; hence there exists a k such that [Equation] Rearrangement of this gives [Equation] This contradicts the supposition that [Equation]. By bisection Divide the interval [Equation] into two halves. One of these halves must contain [Equation]. Select this half, and iterate the process. This creates a sequence of intervals; the end-points constitute sequences of numbers both of which tend to the same limit, which is M.
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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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