Continuity Sec. 2.3.

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Presentation transcript:

Continuity Sec. 2.3

Warm-up 9/3/2013 a. Find the average rate of change on the interval [2 , 5]. b. Find the instantaneous rate of change at x = 2.

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. This function has discontinuities at x=1 and x=2. 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.) Non-Removable (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.

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

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