Continuity and One-sided Limits

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# Continuity and One-sided Limits - PowerPoint PPT Presentation

Continuity and One-sided Limits. A function f is continuous at x = c if there is no interruption in the graph at c The graph is unbroken at c , with no holes, jumps, or gaps A function is continuous at c if f(c) is defined lim f(x) exists lim f(x) = f(c). x  c. x  c.

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Continuity and One-sided Limits
• A function f is continuous at x = c if there is no interruption in the graph at c
• The graph is unbroken at c, with no holes, jumps, or gaps
• A function is continuous at c if
• f(c) is defined
• lim f(x) exists
• lim f(x) = f(c)

x c

x c

Continuity and One-sided Limits
• A function f is continuous on the open interval (a, b), if it is continuous at each point in the open interval
• A function that is continuous on the open interval (-∞,∞) is called everywhere continuous
• If a function f is not continuous at x = c then it is said to have a discontinuity at c
• A discontinuity at c is removable if f can be made continuous by appropriately defining or redefining f(c)
• A discontinuity at c is nonremovable if f cannot be made continuous by appropriately defining or redefining f(c)
Continuity and One-sided Limits
• Which graphs below show a removable discontinuity?
Continuity and One-sided Limits
• Which of these functions are continuous?
Continuity and One-sided Limits
• A one-sided limit is the limit of a function f(x) as x approaches c from just the left or just the right
• lim f(x) = L is the limit as x approaches c from values greater than c
• lim f(x) = L is the limit as x approaches c from values less than c
• One-sided limits can be useful when finding limits of functions involving radicals

Theorem 1.10 Existence of a Limit

If f is a function and c and L are real numbers, then lim f(x) = L if and only if lim f(x) = L and lim f(x) = L

x c+

x c–

x c

x c–

x c+

Continuity and One-sided Limits
• If a function f is continuous on the open interval (a, b), and lim f(x) = f(a) and lim f(x) = f(b), then the function is continuous on the closed interval [a, b]
• The function is said to be continuous from the right at a and continuous from the left at b

x a+

x b–

Continuity and One-sided Limits

Theorem 1.11 Properties of Continuity

If b is a real number and f and g are functions that are continuous at c, then the following functions are also continuous at c

• Scalar multiple: bf
• Sum or difference: f ± g
• Product: f · g
• Quotient: f/g provided that g(c) ≠ 0
Continuity and One-sided Limits
• These properties imply that
• Polynomial functions are continuous at every real number
• Rational, radical, and trigonometric functions are continuous at every number in their domain

Theorem 1.12 Continuity of Composite Functions

If g is continuous at c and f is continuous at g(c), then the composite function f◦g(x) = f(g(x)) is continuous at c

Intermediate Value Theorem

Theorem 1.13 Intermediate Value Theorem

If f is continuous on the closed interval [a, b], f(a) ≠ f(b), and k is any real number between f(a) and f(b), then there is at least one number c in [a, b] such that f(c) = k

• This means that for a continuous function f, if x takes on all values between a and b, then f(x) must take on all values between f(a) and f(b)
• The Intermediate Value Theorem is an existence theorem, it tells us that c exists, but not its value
Intermediate Value Theorem
• The Intermediate Value Theorem guarantees that for a continuous function f there is at least one number c in the closed interval [a, b] such that f(c) = k
• It’s possible that there may be more than one number cin [a, b], such that f(c) = k
• If f is not continuous theintermediate value theorem does not apply, and there may be no number c in [a, b]such that f(c) = k
Intermediate Value Theorem
• The Intermediate Value Theorem can be used to find the zeros of a function that is continuous on a closed interval
• If f is continuous on [a, b] andf(a) and f(b) have different signs, then there must be some value cin [a, b], such that f(c) = 0
• This gives the bisection method for approximating the real zeros of a continuous function