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Understanding Continuity and Discontinuity in Mathematics

Learn about discontinuities in functions, types of discontinuity, one-sided limits, formal definition of continuity, and examples of calculating limits and discontinuities. Explore concepts of removable and non-removable discontinuities.

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Understanding Continuity and Discontinuity in Mathematics

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  1. Section 2.4 Continuity & One-Sided Limits

  2. Discontinuous v. Continuous

  3. Formal Definition of Continuity

  4. Two Types of Discontinuity • Removable & Non-Removable • A discontinuity at is called removable if can be made continuous by appropriately defining (or redefining)

  5. One-Sided Limits • “the limit of as approaches from the right” • “the limit of as approaches from the left”

  6. When Does a Limit Exist?

  7. Example 1 (#6)

  8. Calculating One-Sided Limits • Plug in the x-value that you are approaching. • If you get a real number, then that’s your limit. • If not, try some algebra to see if things can cancel. • If that doesn’t work, then plug in x-values extremely close to the number you are approaching or graph it.

  9. Example 2 Find the limit if it exists. If not, explain.

  10. Example 2 (cont.) Find the limit if it exists. If not, explain. • where • where

  11. Continuity on a Closed Interval

  12. Pictorial Representation

  13. Example 3 Discuss the continuity of the function on the closed interval.

  14. Example 4 Find the -values where is discontinuous. If any, which are removable?

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