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Continuity in Mathematics: Types and Examples

Learn about continuity in functions, types of discontinuities, and the Intermediate Value Theorem. Explore examples and explanations to grasp this fundamental concept in mathematics.

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Continuity in Mathematics: Types and Examples

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  1. 1.4 Continuity • f is continuous at a if • is defined. • exists.

  2. Ex 1: Discontinuous where & why? *see graph.

  3. 1.4 Continuity • 3 types of discontinuity: • Removable • Infinite • Jump

  4. Ex 2: Discontinuous where & why?

  5. Ex 2: Discontinuous where & why?

  6. Functions are continuous at every number in their domains!

  7. Continuity on a Closed Interval • f is continuous on [a,b] if it is continuous on (a, b) and:

  8. Ex 3: Show that f(x) is continuous on the interval [1, 1]

  9. Ex 4: Continuous where?

  10. The Intermediate Value Theorem (IVT): If f is continuous on the interval [a, b] and k is any number between f(a) & f(b), then there exists a number c in (a, b) such that f(c) = k.

  11. Ex 5: Show that the equation has a root in the interval [1, 2]

  12. 1.4 pg. 781 – 5 odds,7 – 23 EOO,25 – 31 odds,33 – 53 EOO,57, 59, 75, 77, 8523 Total

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