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2.3 continuity

2.3 continuity. Definition: Continuity At A Point. Interior point: a function is continuous at an interior point c of its domain if . Endpoint: a function is continuous at a left endpoint a or is continuous at a right endpoint b of its domain if or respectively.

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2.3 continuity

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  1. 2.3 continuity

  2. Definition: Continuity At A Point • Interior point: a function is continuous at an interior point c of its domain if . • Endpoint: a function is continuous at a left endpoint a or is continuous at a right endpoint b of its domain if or respectively.

  3. Checklist For Continuity: • exists • exists

  4. Continuous function • A continuous function is one that is continuous at every point of its domain.

  5. Examples: 1. 2. • graph each function • Find the domain of each • Is it a continuous function?

  6. Removing a Discontinuity: • Let • What is the domain of ? • Find all discontinuities. • Give a formula for the extended function that is continuous at all removable discontinuities.

  7. Theorem 6: Properties of Continuous Functions • If the functions are continuous at , then the following combinations are continuous at . • , for any number

  8. Theorem 7: Composite Of Continuous Functions • If is continuous at and is continuous at , then the composite is continuous at

  9. Theorem 8: Intermediate Value Theorem for Continuous Functions • A function that is continuous on a closed interval takes on every value between and . In other words, if is between and , then for some in .

  10. Example: • Is any real number exactly 7 less than its cube?

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