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Understanding Continuous Functions in Calculus

Learn how to identify and remove discontinuities, apply Intermediate Value Theorem, and solve problems involving continuous functions. Homework exercises included.

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Understanding Continuous Functions in Calculus

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  1. Today in Calculus • Notes • Continuous Functions • Identifying types of discontinuity • Removing removable discontinuities • Intermediate Value Theorem • Go over quiz • Homework

  2. Continuous Functions Definition • Interior point: A function is continuous at an interior point c of its domain if • Endpoint: A function is continuous at a left endpoint a or right endpoint b of its domain if If a function is undefined at a point, then it’s discontinuous at that point. a b

  3. Removable Discontinuity

  4. Oscillating Discontinuity It oscillates too much to have a limit as x→0

  5. Jump Discontinuity

  6. Infinite Discontinuity

  7. Discontinuities

  8. Finding points of discontinuity Where does the function not exist? • Find VA and holes • Use graph or table

  9. Removing a removale discontinuity • Factor and find holes • Find limit at each removable point, set

  10. Removing a discontinuity • Is the function continuous at x=1? • What value should be assigned to f(1) to make the extended function continuous at x = 1?

  11. Example Determine the value of b for which f(x) is continuous at x = 2.

  12. 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), the y0 = f(c) for some c in [a, b]. Example: f(x) is continuous and has exactly one zero. If f(-3)=4 and f(2)=-5, At which value of x does f(x)=0? a) -7 b) -4 c) 1 d) 3 e)4

  13. Homework • Pg 80: 1,7,9,11-16,21,24,25,26,35-38 • Bring calculators tomorrow

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