1. 2.3 Continuity Life is a continuity always and always.There is no final destination it is going towards.Just the pilgrimage, just the journey in itself is life,not reaching to some point,no goal - just dancing and being in pilgrimage,moving joyously, without bothering about any destination.- Osho

2. 2 1 1 2 3 4 Most of the techniques of calculus require that functions be continuous. A function is continuous if you can draw it in one motion without picking up your pencil. A function is continuous at a point if the limit is the same as the value of the function. This function has discontinuities at x=1 and x=2. It is continuous at x=0 and x=4, because the one-sided limits match the value of the function

3. Removable Discontinuities: (You can fill the hole.) Essential Discontinuities: oscillating infinite jump

4. has a discontinuity at . Write an extended function that is continuous at . Note: There is another discontinuity at that can not be removed. Removing a discontinuity:

5. Note: There is another discontinuity at that can not be removed. Removing a discontinuity:

6. Also: Composites of continuous functions are continuous. Continuous functions can be added, subtracted, multiplied, divided and multiplied by a constant, and the new function remains continuous. examples:

7. Because the function is continuous, it must take on every y value between and . Intermediate Value Theorem If a function is continuous between a and b, then it takes on every value between and .

8. F2 Since f is a continuous function, by the intermediate value theorem it must take on every value between -1 and 5. Therefore there must be at least one solution between 1 and 2. Use your calculator to find an approximate solution. Is any real number exactly one less than its cube? Example 5: (Note that this doesn’t ask what the number is, only if it exists.) 1: solve

9. Graph: Note resolution. Graphing calculators can sometimes make non-continuous functions appear continuous. CATALOG F floor( This example was graphed on the classic TI-89. You can not change the resolution on the Titanium Edition. The calculator “connects the dots” which covers up the discontinuities.

10. Graph: GRAPH Graphing calculators can make non-continuous functions appear continuous. CATALOG F floor( If we change the plot style to “dot” and the resolution to 1, then we get a graph that is closer to the correct floor graph. The open and closed circles do not show, but we can see the discontinuities. p

11. Yes! Its your Fun and Happy Joy Joy Pleasure Time #3-30 (multiples of 3), 36, 39, 42, 43, 48 Write #45 as a Journal.