Continuity
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Continuity
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Intuitively, a function is continuous at a point if, when we draw its graph at that point we can do so without lifting the pen. The formal definition of continuity is based on the notion of a limit. In what follows the phrase “from above” means that we replace the condition [Equation goes here - download the original to see it.] in the definition of a limit by [Equation]. That is a function [Equation] tends to the limit l from above as x gets closer and closer to a if, when e is a given positive number, however small, a number can be found, depending on e such that [Equation] This is abbreviated to[Equation] The phrase “from below” means that we replace [Equation] in the definition of a limit by [Equation]. This is abbreviated to[Equation] Continuity. The function [Equation] is continuous when [Equation] if [Equation] tends to a limit l as x tends to a from above and to the same limit l as x tends to a from below, while [Equation] when [Equation]. In conclusion, to show that a function is continuous at a point a we have to show [Equation] Equivalent definition. The function [Equation] is continuous when [Equation] if [Equation]. That is, there is a [Equation] such that [Equation]. Neighbourhood. The open set [Equation] is called a neighbourhood of a. A function f is said to be continuous in an open interval if it is continuous at every point of the interval. It is continuous in a closed interval [Equation] if it is continuous at all points contained within the interval and if [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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