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- 1. 1.4 Continuity and One-Sided Limits
- 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>
- 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.
- 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?
- 7. Domain: ? Is it continuous at every x-value in its domain? Is there any way to fill the discontinuity?
- 8. Domain: ? Is it continuous at every x-value in its domain? Notice as x ->0, the limit is 1, so they connect
- 9. Domain: ? Is it continuous at every x-value in its domain?
- 10. One-Sided Limits and Continuity on a closed interval
- 11. Ex 2 p. 72 A One-Sided Limit
- 12. Ex 3 p. 72 Greatest Integer Function
- 13. When limit from left is not equal to limit from right, the (two-sided) limit does not exist
- 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.
- 15. Ex 4 p. 73 Continuity on a Closed Interval Discuss continuity of Domain is [-1, 1] (closed) Continuous on interior Is it continuous?
- 17. Ex 6, p.75 Applying Properties of Continuity Are these functions continuous? Why or why not?
- 18. Think about inside function and outside.
- 19. Ex 7, p. 76 Testing for Continuity Describe the interval(s) on which function is continuous.
- 20. Assignment p. 78/ 1-69 EOO

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