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Continuity

Continuity. TS: Making decisions after reflection and review. Objectives. To find the intervals on which a function is continuous. To find any discontinuities of a function. To determine whether discontinuities are removable or non-removable. Video Clip from Calculus-Help.com. Continuity.

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Continuity

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  1. Continuity TS: Making decisions after reflection and review

  2. Objectives • To find the intervals on which a function is continuous. • To find any discontinuities of a function. • To determine whether discontinuities are removable or non-removable.

  3. Video Clip fromCalculus-Help.com Continuity

  4. What makes a function continuous? • Continuous functions are predictable… 1) No breaks in the graph A limit must exist at every x-value or the graph will break. 2) No holes or jumps The function cannot have undefined points or vertical asymptotes.

  5. Continuity • Key Point: Continuous functions can be drawn with a single, unbroken pencil stroke.

  6. Continuity • Mathematically speaking… If f (x) is continuous, then for every x = c in the function, • In other words, if you can evaluate any limit on the function using only the substitution method, then the function is continuous.

  7. Continuity of Polynomial and Rational Functions • A polynomial function is continuous at every real number. • A rational function is continuous at every real number in its domain.

  8. Polynomial Functions • Both functions are continuous on .

  9. Rational Functions continuous on: continuous on:

  10. Rational Functions continuous on: continuous on:

  11. Piecewise Functions continuous on

  12. Discontinuity • Discontinuity: a point at which a function is not continuous

  13. Discontinuity • Two Types of Discontinuities 1) Removable (hole in the graph) 2) Non-removable (break or vertical asymptote) • A discontinuity is calledremovable if a function can be made continuous by defining (or redefining) a point.

  14. Two Types of Discontinuities

  15. Find the intervals on which these function are continuous. Discontinuity Point of discontinuity: Removable discontinuity Vertical Asymptote: Non-removable discontinuity

  16. Discontinuity Continuous on:

  17. Discontinuity Continuous on:

  18. Discontinuity • Determine the value(s) of x at which the function is discontinuous. Describe the discontinuity as removable or non-removable. (A) (B) (C) (D)

  19. Discontinuity (A) Removable discontinuity Non-removable discontinuity

  20. Discontinuity (B) Removable discontinuity Non-removable discontinuity

  21. Discontinuity (C) Removable discontinuity Non-removable discontinuity

  22. Discontinuity (D) Removable discontinuity Non-removable discontinuity

  23. Conclusion • Continuous functions have no breaks, no holes, and no jumps. • If you can evaluate any limit on the function using only the substitution method, then the function is continuous.

  24. Conclusion • A discontinuity is a point at which a function is not continuous. • Two types of discontinuities • Removable (hole in the graph) • Non-removable (break or verticalasymptote)

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