2.4 Continuity and its Consequences Fri Sept 16

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2.4 Continuity and its ConsequencesFri Sept 16

• Do Now
• Find the errors in the following and explain why
  its wrong

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HW Review p.80 5 13 19 27 29 33 35

• 5) 1/2
• 13) -2/5
• 19) 1/5
• 27) 3
• 29) 1/16
• 33) let f(x) 1/x and g(x) -1/x
• 35) proof

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Continuity - What does it mean?

• A function is said to be continuous on an
  interval if its graph on that interval can be
  drawn without interruption, or without lifting
  your pencil.
• Holes and asymptotes are examples of
  discontinuous functions

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Definition of continuous

• A function f is continuous at x a when
• 1) f(a) is defined
• 2) exists
• 3)
• Otherwise, f is said to be discontinuous at x a

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One-Sided Continuity

• A function f(x) is called
• Left-continuous at x c if
• Right-continuous at x c if

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What kind of functions are continuous?

• Polynomials
• Radical Functions on their domains
• Sin x and cos x
• Exponential functions
• Logarithmic functions on their domains
• Rational functions on their domains

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Piecewise Functions

• These kind of functions are the big AP type of
  problems

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More Continuous Functions

• Thm- Suppose that f and g are continuous at x
  c. Then
• 1) kf(x) for any constant k
• 2) is continuous at x c
• 3) is continuous at x c
• 4) is continuous at x c if
• and discontinuous if g(c) 0

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More Continuous Functions

• Thm- If f(x) is continuous on an interval I with
  range R and its inverse exists, then its inverse
  is continuous with domain R

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Composite Functions

• If g(x) is continuous at x c, and f(x) is
  continuous at x g(c), then f(g(x)) is also
  continuous at x c

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3 Types of Discontinuities

• Removable Discontinuity
• Limit exists
• F(x) is not equal to the limit
• Can redefine function at discontinuity
• Jump Discontinuity
• Left and right side limits do not agree
• Cannot redefine
• Infinite Discontinuity
• One or both of each sided limits is infinite

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Closure

• Journal Entry What must be true for a function
  to be continuous? What is an example of a
  discontinuity? Which are removable or not?
• HW p.88-89 1, 3-5, 17-33 odds, 55 57 59 63 65

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Continuity ContdMon Sept 19

• Do Now
• Is the function
• continuous at the following points?
• X 3
• X 4

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HW Review p.88-89 1, 3-5, 17-33 odds, 55 57 59
63 65
1) RC_at_1 nether_at_3 LC_at_5 27) x 2, jump, LC 3) X
3 redefine g(3) 4 29) t (2n1)pi/4, n
int 4) C 1, redefine g(1) 3 31) continuous
for all 5) Omgicantfitthishere 33) x 0, inf,
neither 17) X 0, inf, neither 55) show right
lim left 19) X 1, inf, neither 57) c
5/3 21) Even ints, jump, RC 59) a 2, b 1 23)
X 1/2, inf, neither 63) graph 25) Continuous
for all x 65) graph
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Classwork

• Side 1(p.53) 3, 4
• Side 2(p.153) 21 22 23 24 25

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Closure

• Exit pass Find all discontinuities of
• For each discontinuity, state the type, whether
  it is left/right continuous, and if removable,
  redefine it so it is continuous
• HW none or finish worksheet
• 2.3-2.5 quiz soon