Calc 1.4a

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Calc 1.4a

  1. 1. 1.4 Continuity and One-Sided Limits
  2. 2. Continuity at a Point and on an Open Interval <ul><li>Informal definition of continuity: To say a function f is continuous at x = c means there is no interruption at x=c; no holes, no jumps, no gaps. </li></ul>
  3. 4. Discontinuities come in two types – removable and nonremovable. . . .Removable if f can be made continuous by appropriately defining or redefining f(c) , that is, by filling the hole. . . .Nonremovable if there is an asymptote or gap, not just a hole. If a function f is defined on an open interval I (except possibly at x=c) and f is not continuous at c, then f has a discontinuity.
  4. 6. Ex 1, p. 71 Continuity of a Function Discuss continuity: Domain: ? Is it continuous at every x-value in its domain? Is there any way to fill the discontinuity?
  5. 7. Domain: ? Is it continuous at every x-value in its domain? Is there any way to fill the discontinuity?
  6. 8. Domain: ? Is it continuous at every x-value in its domain? Notice as x ->0, the limit is 1, so they connect
  7. 9. Domain: ? Is it continuous at every x-value in its domain?
  8. 10. One-Sided Limits and Continuity on a closed interval
  9. 11. Ex 2 p. 72 A One-Sided Limit
  10. 12. Ex 3 p. 72 Greatest Integer Function
  11. 13. When limit from left is not equal to limit from right, the (two-sided) limit does not exist
  12. 14. Idea of one-sided limit extends definition of continuity to closed intervals. It is continuous on closed intervals if it is continuous in the interior and exhibits one-sided continuity at the endpoints.
  13. 15. Ex 4 p. 73 Continuity on a Closed Interval Discuss continuity of Domain is [-1, 1] (closed) Continuous on interior Is it continuous?
  14. 17. Ex 6, p.75 Applying Properties of Continuity Are these functions continuous? Why or why not?
  15. 18. Think about inside function and outside.
  16. 19. Ex 7, p. 76 Testing for Continuity Describe the interval(s) on which function is continuous.
  17. 20. Assignment p. 78/ 1-69 EOO

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