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lecture04 - Lecture 4 Continuity and Exponential Functions...

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Lecture 4 - Continuity and Exponential Functions 4.1 Continuity Recall that if f ( x ) is a polynomial, then lim x c f ( x ) = f ( c ). These types of limits are easy to calculate. This leads to the following definition. Definition 4.1 Let c ( a, b ) and f ( x ) a function whose domain contains ( a, b ) . then the function f ( x ) is continuous at c if lim x c f ( x ) = f ( c ) . Note that this implies 1. f ( c ) is defined, 2. the limit exists, and 3. the two are equal. Intuition: The graph of a continuous function is one that has no holes, jumps, or gaps. It can be “drawn without lifting the pencil”. This is intuition only . Example: f ( x ) is not continuous at 1. x = 1 because f (1) is not defined.
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2. x = 2 because lim x 2 does not exist. 3. x = 4. Here the limit exists, but is not equal to f (4). These are the three basic ways something can fail continuity. Examples: 1. Any polynomial p ( x ) is continuous everywhere. 2. A rational function is one of the form f ( x ) = p ( x ) q ( x ) where p ( x ) and q ( x ) are polynomials. If f ( x ) is a rational function, it will be continuous everywhere except where q ( x ) = 0 (in these places, f ( x ) is undefined, hence certainly not continuous). In general, if f ( x ) = p ( x ) q ( x ) , where p and q are arbitrary, then f ( x ) is continuous every- where that p and q are continuous and q is not 0. 3. Consider f ( x ) = x 2 - 1 x - 1 . Let’s graph it. f ( x ) is undefined at x = 1. If x 6 = 1, we get x 2 - 1 x - 1 = ( x + 1)( x - 1) x - 1 = x + 1 – this is a line.
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