Functions and Continuity
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Continuity
We will now explicitly define continuity. In this definition the phrase "from above" means that we replace the condition [Equation goes here - download the original pdf to see it.] in the definition of a limit by [Equation goes here - download the original pdf to see it.]. That is A function [Equation goes here - download the original pdf to see it.] 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 [symbol] can be found, depending on e such that [Equation goes here - download the original pdf to see it.] This is abbreviated to[Equation goes here - download the original pdf to see it.] The phrase "from below" means that we replace [Equation goes here - download the original pdf to see it.] in the definition of a limit by [Equation goes here - download the original pdf to see it.] . This is abbreviated to[Equation goes here - download the original pdf to see it.] The function [Equation goes here - download the original pdf to see it.] is continuous when [Equation goes here - download the original pdf to see it.] if [Equation goes here - download the original pdf to see it.] 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 goes here - download the original pdf to see it.] when [Equation goes here - download the original pdf to see it.]. To show that a function is continuous at a point a we have to show [Equation goes here - download the original pdf to see it.]
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Contents of
Functions and Continuity
1 Functions
2 Graph
3 Inverse of a Function: monotone increasing or decreasing functions
4 The reciprocal function - singularities
5 Functions defined piecewise on their domain
6 Limits
7 Formal definition of a limit
8 Formal definition of the limit of a function at a point x = a
9 Informal arguments
10 Continuity
11 Combining limits
12 Quotients
13 Image set
14 Inverse image
15 Odd and even functions
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