Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 2.3 Continuity
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide What you’ll learn about Continuity at a Point Continuous Functions Algebraic Combinations Composites Intermediate Value Theorem for Continuous Functions …and why Continuous functions are used to describe how a body moves through space and how the speed of a chemical reaction changes with time.
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Continuity at a Point
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Continuity at a Point If a function f is not continuous at a point c, we say that f is discontinuous at c and c is a point of discontinuity of f. Note that c need not be in the domain of f.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Continuity at a Point The typical discontinuity types are: a)Removable(2.21b and 2.21c) b)Jump(2.21d) c)Infinite(2.21e) d)Oscillating (2.21f)
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Continuous Functions A function is continuous on an interval if and only if it is continuous at every point of the interval. A continuous function is one that is continuous at every point of its domain. A continuous function need not be continuous on every interval.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Continuous Functions [-5,5] by [-5,10]
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Intermediate Value Theorem for Continuous Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide Intermediate Value Theorem for Continuous Functions The Intermediate Value Theorem for Continuous Functions is the reason why the graph of a function continuous on an interval cannot have any breaks. The graph will be connected, a single, unbroken curve. It will not have jumps or separate branches.