chapter2(4) - MATH1010 Chapter 2 cont 1 LIMITS AND...

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MATH1010: Chapter 2 cont… 1 LIMITS AND DERIVATIVES cont… Continuity (Section 2.5 of Stewart, pg. 124) Question: How does = + = 3 3 2 1 ) ( x x x x f compare with 1 ) ( + = x x g ? Definition: A function f is continuous at a number a if ) ( ) ( lim a f x f a x = So, what actually has to hold? Graphical Example: What happens if we are given a formula for the function?…how do we determine where the function is continuous? Example: 1 4 5 ) ( 2 + + + = x x x x f
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MATH1010: Chapter 2 cont… 2 Example: = = 3 3 6 3 1 ) ( x x x x f Application: Suppose that a certain country has the following income tax rates, where x is income in thousands of dollars. > < = 20 20 10 10 15 . 0 1 . 0 0 ) ( x x x x f Definition: A function f is continuous from the right at a number a if ) ( ) ( lim a f x f a x = + and f is continuous from the left at a if ) ( ) ( lim a f x f a x = Graphical Example: Example: > + + = 0 0 7 2 1 4 ) ( x x x x x f So far, we’ve only talked about continuity at a point, but what do we mean when we say a function is continuous on e.g. [0, 6]?
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chapter2(4) - MATH1010 Chapter 2 cont 1 LIMITS AND...

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