2 3 Continuity Grand Canyon Arizona Photo by

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

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. 2 1 1 2 3 4 It is continuous at x=0 and x=4, because the one-sided limits match the value of the function

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

Where have we seen those types of discontinuities before? A. They were in a movie we watched in class. B. They were topics of Mc. Kenzie’s bad jokes. C. They were the reasons for limits not existing. D. They were the names of three of the seven dwarfs.

We said this function has a ‘removable discontinuity’, how can we also describe it with ‘Calculus terms”? A. It has a derivative at a. B. It has a limit at a. C. The integral exists at a. D. It is continuous at a. a

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

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

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

If we have a continuous function that has f(3)=-2 and f(4)=4, which of these MUST be true? A. f(5) must be positive B. f(2) must be negative C. f(x)=0 must have a solution between 3 and 4 D. f(x) must be a linear equation with a slope of 6 E. None of the above

Example 5: Is any real number exactly one less than its cube? (Note that this doesn’t ask what the number is, only if it exists. ) 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. F 2 1: solve

Graphing calculators can sometimes make noncontinuous functions appear continuous. Graph: CATALOG Note resolution. 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.

Graphing calculators can make non-continuous functions appear continuous. Graph: 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 GRAPH discontinuities.