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Continuity

Continuity. 2.3. Point Continuity. There are three tests that must occur in order for a function to be continuous at a point c . 1. exists at 2. exists as AND! 3. Continuity at a Point.

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Continuity

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  1. Continuity 2.3

  2. Point Continuity • There are three tests that must occur in order for a function to be continuous at a point c. • 1. exists at • 2. exists as AND! • 3.

  3. Continuity at a Point Any function whose graph can be sketched in one continuous motion is ONE example of a continuous function.

  4. Example:Continuity at a Point o

  5. Continuity at a Point

  6. Continuity at a Point If a function f is not continuous at a point c , we say that f is discontinuous at c and c is a point of discontinuity of f. Note that c need not be in the domain of f.

  7. Types of Discontinuities • point (removeable) • jump • infinite • oscillating

  8. 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

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

  10. 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:

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

  12. Example Continuity at a Point [-5,5] by [-5,10]

  13. Properties of Continuous Functions

  14. 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:

  15. Continuous Function A function is continuous on an interval if and only if it is continuous at every point in the interval. A continuous function is one that is continuous at every point of its domain.

  16. Intermediate Value Theorem A function y = f(x) that is continuous on a closed interval [a,b] takes on every value between f(a) and f(b). In other words, if y0 is between f(a) and f(b), then y0 = f(c) for some c in [a,b]

  17. 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 .

  18. Intermediate Value Theorem for Continuous Functions The Intermediate Value Theorem for Continuous Functions is the reason why the graph of a function continuous on an interval cannot have any breaks. The graph will be a connected, a single, unbroken curve. It will not have jumps or separate branches.

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