2311lectureoutline2.5 - continuous on an interval if it is...

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MAC2311, Calculus I 2.5 Continuity Definition: A function f(x) is continuous at a number a if ( ) ( ) lim x a f x f a = . So, if f(x) is continuous at a , then: 1. f(a) is defined (that is, a is in the domain of f) 2. ( ) lim x a f x exists 3. ( ) ( ) lim x a f x f a = If f(x) is not continuous at a , we say that f(x) has a discontinuity at a . Removable discontinuities occur where there are “holes” in the graph of f(x). Non-removable discontinuities occur where there are vertical asymptotes or jumps in the graph of f(x). Definition: A function f(x) is continuous from the right at a number a if ( ) ( ) lim x a f x f a + = , and f(x) is continuous from the left at a if _ ( ) ( ) lim x a f x f a = . Definition: A function f(x) is continuous on an interval
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Unformatted text preview: continuous on an interval if it is continuous at every number in the interval. (If f(x) is defined only on one side of an endpoint of the interval, we understand continuous at the endpoint to mean continuous from the right or continuous from the left.) Theorem: The following types of functions are continuous at every number in their domains: polynomials, rationals, roots, trigonometric, inverse trigonometric, exponential, logarithmic. The Intermediate Value Theorem: Let f(x) be continuous on the closed, bounded interval [a, b] , and let N be any number between f(a) and f(b), where f(a) ≠ f(b). Then, there exists a number c between a and b , such that f(c) = N . Try Exercises 4, 10, 18, 22, 24, 32, 36, 50 on pages 128-129 in the textbook....
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