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Continuity

Continuity. 2.4. 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.

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Continuity

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  1. Continuity 2.4

  2. 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

  3. Show g(x)=x^2 + 1 is continuous at x = 1

  4. Types of Discontinuities • There are 4 types of discontinuities • Jump • Point • Essential • Removable • The first three are considered non removable

  5. Jump Discontinuity • Occurs when the curve breaks at a particular point and starts somewhere else • Right hand limit does not equal left hand limit

  6. Point Discontinuity • Occurs when the curve has a “hole” because the function has a value that is off the curve at that point. • Limit of f as x approaches x does not equal f(x)

  7. Essential Discontinuity • Occurs when curve has a vertical asymptote • Limit dne due to asymptote

  8. Removable Discontinuity • Occurs when you have a rational expression with common factors in the numerator and denominator. Because these factors can be cancelled, the discontinuity is removable.

  9. Places to test for continuity • Rational Expression • Values that make denominator = 0 • Piecewise Functions • Changes in interval • Absolute Value Functions • Use piecewise definition and test changes in interval • Step Functions • Test jumps from 1 step to next.

  10. Continuous Functions in their domains • Polynomials • Rational f(x)/g(x) if g(x) ≠0 • Radical • trig functions

  11. Find and identify and points of discontinuity Non removable – jump discontinuity

  12. Find and identify and points of discontinuity Non removable – essential discontinuity VA at x = 4

  13. Find and identify and points of discontinuity 2 points of disc. (where denominator = 0) Removable disc. At x = 5 Non removable essential at x = -1 (VA at x = -1)

  14. Find and identify and points of discontinuity Non removable point discontinuity

  15. Find and identify and points of discontinuity 2 points of disc. (where denominator = 0) Removable disc. At x = 5 Non removable essential at x = -4 (VA at x = -4)

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