1'4 Continuity and onesided limits

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1.4 Continuity and one-sided limits

• "Mathematics -- the subtle fine art." -- Jamie
  Byrnie Shaw

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Objective

• To determine the continuity of functions
• To find one-sided limits

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Top 10 excuses for not doing your math homework

• 10. Galileo didn't know calculus what do I need
  it for?
• 9. "A math addict stole my homework.
• 8. I'm taking physics and the homework in there
  seemed to involve math, so I thought I could just
  do that instead.
• 7. I have the proof, but there isn't room to
  write it in the margin.
• 6. I have a solar powered calculator and it was
  cloudy.

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Cont

• 5. I was watching the World Series and got tied
  up trying to prove that it converged.
• 4. I could only get arbitrarily close to my
  textbook. I couldn't actually reach it. (I
  reached half way, and then half of that, and then
  ...)
• 3. I couldn't figure out whether i am the square
  root of negative one or i is the square root of
  negative one.
• 2. It was Einstein's birthday and pi day and we
  had this big celebration! (This only works for
  March 14)
• 1. I accidentally divided by zero and my paper
  burst into flames.

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Intuitive approach
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Formal definition

• A function is continuous at c if
• 1. f(c) is defined ( f(c) exists )
• 2.
• 3.

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Simply put..

• A function is continuous if you can draw it
  without picking up your pencil
• Can be continuous over open or closed intervals,
  or the entire function

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Examples

• When are the following functions continuous?

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Discontinuities

• If a function is discontinuous at a point, the
  discontinuity may be removable or non-removable
  depending upon whether the limit of the function
  exists at the point of discontinuity
• In other words
• Removable holes
• Non-removable- breaks or asymptotes

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Removable discontinuities

• Denominators of fractions that factor with the
  numerators
• Ex
• Holes occur where you can take the limit but the
  actual value does not exist

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Non-removable discontinuities

• Asymptotes- when the denominator is zero
• Ex
• This is when the actual value and the expected
  value or limit do not exist

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Properties of continuity

• If b is a real number, and f and g are continuous
  at x c, then
• 1. Scalar multiple bf(x) is continuous at c
• 2. Sum and difference f g is continuous at c
• 3. Product fg is continuous at c
• 4. Quotient f/g is continuous at c, as long as
  g(c) does not equal 0

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Left and right limits

• We can look at limits from just the left or just
  the right
• Right Left

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Greatest integer function

• Graph
• Find

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Remember

• Three things must happen for a limit to exist
• The limit from the right exists
• The limit from the left exists
• The limit from the right equals the limit from
  the left

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Intermediate Value Theorem (IVT)

• If f is continuous on a,b and k is a number
  between f(a) and f(b) then there is a c such that
  if a lt clt b then f(c) k