2.4 Continuity Grand Canyon, Arizona

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

This function has discontinuities at x=1 and x=2. 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. 1 2 3 4 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

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

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 .

Is any real number exactly one less than its cube? 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. F2 1: solve

Graphing calculators can make non-continuous functions appear continuous. CATALOG F floor( Note resolution. The calculator “connects the dots” which covers up the discontinuities.

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. GRAPH p