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Continuity

Continuity. What is continuity? Algebraic properties Compositions. Definition. A function f is continuous at x=c if f(c) is defined lim x c f(x) exits lim x c f(x) = f(c) If one of these fails to hold, say that f is discontinuous at c. Examples.

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Continuity

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  1. Continuity • What is continuity? • Algebraic properties • Compositions

  2. Definition • A function f is continuous at x=c if • f(c) is defined • limxc f(x) exits • limxc f(x) = f(c) • If one of these fails to hold, say that f is discontinuous at c.

  3. Examples • The absolute value function is continuous everywhere • Polynomials are continuous • Which of the following are continuous at x=3?

  4. More Examples

  5. Bracket Function

  6. Right and Left Continuity

  7. Another example

  8. Question

  9. Non-removable discontinuities

  10. Combinations of Continuous Functions

  11. Important facts

  12. Examples • For which x are the following functions continuous?

  13. Compositions • Suppose that limxc g(x)=L and that f is continuous at L. Then limxc f(g(x)) = f(L), i.e. limxc f(g(x)) = f(limxc g(x)) • Consequence: If g is continuous at x=c and if f is continuous at g(c) then f(g(x)) continuous at c (composition of continuous functions is continuous)

  14. Problems

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