Answered: What does it mean for a function to be…

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter3: Functions

Section3.3: More On Functions; Piecewise-defined Functions

Problem 83E

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What does it mean for a function to be continuous on an interval?
Give examples to illustrate the fact that a function that is not continuous on its entire domain?

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$F o r o p e n i n t e r v a l I f f : (a, b) \to ℝ i s d e f i n e d o n a n o p e n i n t e r v a l, t h e n f i s c o n t i n u o u s o n (a, b) i f a n d o n l y i f \lim_{x \to c} f (x) = f (c) w h e r e a < c < b \sin c e e v e r y p o i n t o f (a, b) i s a n a c c u m u l a t i o n p o i n t . F o r c l o s e d i n t e r v a l I f f : [a, b] \to ℝ i s d e f i n e d o n a c l o s e d, b o u n d e d i n t e r v a l, t h e n f i s c o n t i n u o u s o n [a, b] i f a n d o n l y i f \lim_{x \to c} f (x) = f (c) w h e r e a < c < b \lim_{x \to a +} f (x) = f (a), \lim_{x \to b +} f (x) = f (b)$

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