Section 2'5 Continuity

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Section 2.5 Continuity
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Continuity at a Point

• A function is continuous at a number if
  (and only if)

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Three Ramifications

• The statement implies three things
• Limit exists
• Domain includes
• Limit is same as

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Example 2a (removable discontinuity)
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Example 2c
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Example 2b (Infinite discontinuity)
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Example 2d (Jump discontinuities)
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Continuity on (a,b)

• f continuous on (a,b) if and only if it is
  continuous at every number in the interval

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Continuity from the Left

• A function is continuous from the left at a
  number if (and only if)

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Continuity from the Right

• A function is continuous from the right at a
  number if (and only if)

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Continuity on Closed Interval

• f closed on a,b if and only if
• f continuous from right at x a
• f continuous on (a,b)
• f continuous from left at x b

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Example Continuity on -2,2
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Theorem 4, p. 127

• If f, g continuous at a and c is a constant, then
  the following are cont. at a, too
• f g f g
• c f fg
• If g(a)?0, then f / g is cont. at a, too.
• (Discuss proof for f g. Others HW)

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Theorem 5, p. 127

• Polynomials continuous on ( -8, 8)
• Rational functions continuous on domain

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Theorem 7, p. 129

• The following are cont. on their domains

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Example 6

• Determine where the function is continuous

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Discuss Intermediate Value Theorem

• Exercise 48

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HW Assignment

• Section 2.5 1-17 (odd), 26, 30, 31, 35, 36,
  37, 41, 43, 49, 51