Functions and Continuity
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Odd and even functions
The graphs of some functions exhibit symmetry. For example [symbol] is symmetrical about the y-axis. [Diagram goes here - download the original pdf to see it.] Such functions are said to be even or symmetric functions. The formal criterion for an even function is [Equation goes here - download the original pdf to see it.] A function which is not even may yet be "almost" symmetrical. For example, in the graph of [Equation goes here - download the original pdf to see it.] [Diagram goes here - download the original pdf to see it.] If we were to reflect the negative part in the x-axis we would obtain an even function. Diagram goes here - download the original pdf to see it.] Example (12) Prove that the function [Example goes here - download the original pdf to see it.] Functions like [Equation goes here - download the original pdf to see it.] that can be turned into even functions in this way are called odd or anti-symmetric. The formal criterion for an odd function is [Equation goes here - download the original pdf to see it.] Example (13) Prove that the function [Example goes here - download the original pdf to see it.] Functions that do not match either criterion are called neither odd not even.
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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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