suplecture1 - Supplemental Lecture 1 Continuity and Limits...

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Supplemental Lecture 1 Continuity and Limits
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I - 2 Functions Let X and Y be sets, and let f X × Y . That means that f is a set of ordered pairs ( x , y ) such that x X and y Y . If f also satisfies the property that for all x X there exists a unique y Y such that ( x , y ) f , then f is a function .
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I - 3 Function Notation We write f : X Y . X is called the domain of f , and Y is called the range of f . If ( x , y ) f , then we write y = f ( x ) and y is called the image of x under f .
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I - 4 The Absolute Value Function The absolute value function |·| : R R + = [0, ) will play an essential role. For x R , Properties of absolute value: a) | xy | = | x || y | b) | |x | - | y || ≤ |x + y | ≤ | x | + | y | (Triangle Inequality) . 0 0 < - = x x x x x -1 -0.5 0.5 1 0.2 0.4 0.6 0.8 1
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I - 5 Definition of Continuity in One Dimension Let x D and y = f ( x ). f is continuous at x iff for all ε > 0 there exists δ > 0 such that, for all z D , if | x - z | < δ then | y - f ( z )| < ε .
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