Continuity
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Bounds of continuous functions
Equations are omitted for technical reasons - download the original pdf
A continuous function on a closed interval is bounded If f is continuous in the closed interval [Equation goes here - download the original to see it.] then f is bounded in [Equation]. It is essential in this theorem that the interval should be closed. On an open interval the theorem does not hold. For instance, the function [Equation] is continuous on the interval but [Equation]. Remark This theorem is very similar to the intermediate value theorem and is proved similarly. Essentially, since the interval is closed and f is bounded, in connecting a to b (which we can visualise by drawing its graph) values of f cannot become unboundedly large, either positive or negative. The proof, which is somewhat technical in detail, basically mimics this argument by requiring values of f to be in a neighbourhood of values of its endpoints. Proof Let S be the set of numbers [Equation] such that [Equation] is bounded above for [Equation]. Since [Equation], S is not empty. Also, if [Equation] then [Equation]. Therefore, S 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]. Then one of the following holds: [Equation] However, the first two of these lead to contradictions. Firstly, suppose [Equation]. The definition of continuity is: The function [Equation] is continuous when [Equation] if [Equation]. That is, there is a [Equation] such that [Equation] for [Equation]. Hence, since f is continuous there exists a neighbourhood [Equation] of [Equation] that is contained in [Equation] such that [Equation]. Since [Equation], there exists a number K and an [Equation] such that [Equation] for [Equation] and [Equation]. Hence [Equation] for [Equation] Hence [Equation] and [Equation] which contradicts the assumption that [Equation]. By a similar argument the assumption that [Equation] leads to a contradiction. Hence [Equation] Since f is continuous at b, there is a [Equation] such that [Equation] for [Equation]. Since [Equation], there exists a K and an [Equation] such that [Equation] for [Equation] and [Equation]. Hence [Equation] for [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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