Functions and Continuity
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Informal arguments
In this chapter "informal" arguments will suffice. That is, we do not use the formal definition of a limit given above, for the reason that the work involved tends to be nasty and often quite unnecessary. Informal arguments are based on the idea that what is obvious is true. For example, it is obvious when [symbol] is defined by [Equation goes here - download the original pdf to see it.] that [Equation goes here - download the original pdf to see it.] converges on the value 1 from both directions; that is [Equation goes here - download the original pdf to see it.] However, when [Equation goes here - download the original pdf to see it.] is the function [Equation goes here - download the original pdf to see it.] we have [Equation goes here - download the original pdf to see it.] [Diagram goes here - download the original pdf to see it.] Since [Equation goes here - download the original pdf to see it.] this proves informally that h is not continuous at x = 0.
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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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