1. Continuity!!

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5. Definitions Continuity at a point: A function f is continuous at c if the following three conditions are met: Continuity at a point: A function f is continuous at c if the following three conditions are met:  is defined  exists 

6. Definitions Continuity on an open interval: A function is continuous on an open interval if it is continuous at each point in the interval Continuity on an open interval: A function is continuous on an open interval if it is continuous at each point in the interval A function that is continuous on the entire real line is everywhere continuous A function that is continuous on the entire real line is everywhere continuous

7. Definitions A function is continuous on a closed interval if it is continuous on the open interval and A function is continuous on a closed interval if it is continuous on the open interval and and and

8. c y=f(x)y=f(x) y x f(c)f(c) Continuous

9. c y=f(x)y=f(x) y x f(c)f(c) Not continuous

10. c y=f(x)y=f(x) y x

11. c y=f(x)y=f(x) y x f(c)f(c)

12. c y=f(x)y=f(x) y x

13. c y=f(x)y=f(x) y x

14. The THREE requirements for a function to be continuous at x=c … 1. C must be in the domain of the function - you can find f(c ), 2. The right-hand limit must equal the left-hand limit which means that there is a LIMIT at x=c, and 3. AND…

15. How does continuity relate to an Etch-A-Sketch? If a function is continuous, there are  NO holes,  NO jumps, and  NO vertical asymptotes. So, if you draw a continuous function, you should be able to draw it without lifting up your pencil …. just like an E EE Etch A Sketch!

16. Types of Discontinuities…  Removable Discontinuities – can be “repaired” –Hole (factor can be “factored out” of the denominator)  Essential Discontinuities – cannot be “repaired” –Jumps (usually found in piecewise functions) –Asymptotes (can’t remove a factor/problem in the denominator) --- (like 1/x) –Wildly oscillating functions – graph (1/sin(x)) and keep zooming in !