Functions and Continuity
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Functions
A function is a mapping from one set (called the domain) to another set (called the co-domain). That is, a function is a rule taking you from one number to another. Functions can be written explicitly by specifying exactly which numbers are mapped to which. For example[Equation goes here - download the original pdf to see it. The symbol [Equation goes here - download the original pdf to see it.] It is only possible to specify finite functions by such an explicit rule, or mapping diagram. Often we define a function by an implicit rule indicating the process that takes you from a number in the domain to a number in the co-domain. [Equation goes here - download the original pdf to see it.] This is read "f is the function from [equation] such that x maps to [Equation goes here - download the original pdf to see it. ." The symbols [Equation goes here - download the original pdf to see it.] [Equation goes here - download the original pdf to see it.] are interchangeable. For a given application of a rule, the number in the domain is called the argument of the function and the number to which it is mapped by the rule is called its value. For example [Equation goes here - download the original pdf to see it.] Here 1 is the argument and 5 is the value. The image set is the set of all numbers in the co-domain that can be values of the function. In set notation [Equation goes here - download the original pdf to see it.] The image may be equal to or smaller than the codomain. For example [Equation goes here - download the original pdf to see it.] We could have written the image as [Equation goes here - download the original pdf to see it.]. This uses the convention that a square (closed) bracket means that the point next to it is included in the set, and a curve bracket means that the point next to it is not included. The symbol [symbol] is used to denote infinity; as infinity is not a number then it cannot be included in the set, so we use a curved (open) bracket next to 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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