Functions and Continuity
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Graph
The graph of a function f from [Equation goes here - download the original pdf to see it.] [Equation goes here - download the original pdf to see it.] So it is a set of points in the x,y-plane. To a point of x there corresponds a point of y. This is a formal definition of a graph. By this stage the student should be very familiar informally with graphs and used to sketching the graphs of functions. Example (1) Constant functions map every point of the domain to a fixed point in the codomain. Let [Equation goes here - download the original pdf to see it.] (a) Describe as a set the graph of f. (b) Sketch the graph of f. (a)The graph is the set [equation] (b)[Diagram goes here - download the original pdf to see it.] Example (2) Affine functions take the form [Example goes here - download the original pdf to see it.] Example (3) Quadratic functions take the form [Example goes here - download the original pdf to see it.] Analytic methods of sketching graphs Polynomial functions take the form [Equation goes here - download the original pdf to see it.] where [equation] are fixed real numbers.. You should be familiar by this stage with analytic methods of sketching polynomial functions - that is, by using the differential calculus to find their turning points and to classify these as maxima or minima. Example (4) Use analytical methods to sketch the graph of [Equation goes here - download the original pdf to see it.]. [Example 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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