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2.3 Continuity

2.3 Continuity. Photo by Vickie Kelly, 2002. Greg Kelly, Hanford High School, Richland, Washington. Grand Canyon, Arizona. 2. 1. 1. 2. 3. 4.

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2.3 Continuity

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  1. 2.3 Continuity Photo by Vickie Kelly, 2002 Greg Kelly, Hanford High School, Richland, Washington Grand Canyon, Arizona

  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

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