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Continuity and One-sided Limits

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- 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

- 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)

- Which graphs below show a removable discontinuity?

- Which of these functions are continuous?

- 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+

- 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–

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

- 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

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

- 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

- 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