advertisement

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 Removing a discontinuity: x3 1 f x 2 x 1 has a discontinuity at x 1 . Write an extended function that is continuous at x 1 . x 1 x 2 x 1 1 1 1 x3 1 lim lim 2 x 1 x 1 x 1 x 1 x 1 2 x3 1 2 , x 1 f x x 1 3 , x 1 2 3 2 Note: There is another discontinuity at x 1 that can not be removed. Removing a discontinuity: 5 4 3 2 1 -5 -4 -3 -2 -1 0 -1 1 2 3 4 5 -2 -3 -4 -5 x3 1 2 , x 1 f x x 1 3 , x 1 2 Note: There is another discontinuity at x 1 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: y sin x 2 y cos x Intermediate Value Theorem If a function is continuous between a and b, then it takes on every value between f a and f b . f b Because the function is continuous, it must take on every y value between f a and f b . f a a b 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.) f 1 1 x x3 1 0 x3 x 1 f x x3 x 1 f 2 5 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. solve x x 3 1, x F2 1: solve 1.32472 Graphing calculators can sometimes make noncontinuous functions appear continuous. Graph: y floor x 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: y floor x 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 weGRAPH can see the discontinuities. p