lect116_6rev_f11

Lect116_6rev_f11 - Thursday September 22 Lecture 6 Continuous functions(Refers to Section 2.5 in your text Students who have mastered the content

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Thursday September 22 Lecture 6 : Continuous functions. ( Refers to Section 2.5 in your text ) Students who have mastered the content of this lecture know about : The definition of “f is continuous at a”, continuity from the left or right of a, a function being continuous on an interval, the basic properties of continuous functions. Students who have practiced the techniques presented in this lecture will be able to : determine whether a simple function is continuous from the definition or its graph, distinguish between removable, infinite and jump discontinuities , apply the basic properties of continuous functions to determine whether a function is continuous, evaluate the limit of a continuous function or a composition of continuous functions at a given point. 6.1 Definition – A function f ( x ) is continuous at a point a if lim x a f ( x ) = f ( a ). 6.1.1 Note – In order that lim x a f ( x ) = f ( a ) three things must hold true: 1) The number a must be in the domain of f . 2) The limit lim x a f ( x ) must exist, i.e. lim x a f ( x ) must be a number. 3) The limit lim x a f ( x ) must be f ( a ). 6.1.2 Proposition – The function f ( x ) is continuous at a if and only if a belongs to the domain of f and lim x a [ f ( x ) – f ( a )] = 0. holds true. Proof: If and only if means we must prove both “directions” and . Suppose a belongs to the domain of f and lim x a [ f ( x ) – f ( a )] = 0 holds true. Then f ( a ) = k is a number. Suppose a belongs to the domain of f and lim x a f ( x ) = f ( a ) is a number.
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6.1.3 Example – The following function g ( t ) is not continuous at t = 0 since lim x 0 g ( t ) = 4 while lim x 0 + g ( t ) = 1.3 and so does lim x 0 g ( t ) not exist. This can be seen in the following graph : 6.1.4 Definition – Suppose f is defined on an interval containing a in the domain of the f . If f is not continuous at a then we say that
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This note was uploaded on 12/28/2011 for the course MATH 116 taught by Professor Robertandre during the Spring '11 term at Waterloo.

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Lect116_6rev_f11 - Thursday September 22 Lecture 6 Continuous functions(Refers to Section 2.5 in your text Students who have mastered the content

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