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Continuity and One-Sided Limits

- Lesson 2.4

(No Transcript)

(No Transcript)

Don't let this Happen to you!

Intuitive Look at Continuity

- A function withoutbreaks orjumps
- The graph can bedrawn without lifting the pencil

?

Continuity at a Point

- A function can be discontinuous at a point
- A hole in the function and the function not

defined at that point - A hole in the function, but the function is

defined at that point

Continuity at a Point

- A function can be discontinuous at a point
- The function jumps to a different value at a

point - The function goes to infinity at one or both

sides of the point, known as a pole

Definition of Continuity at a Point

- A function is continuous at a point x c if the

following three conditions are met - f(c) is defined
- For this link, determine which of the conditions

is violated in the examples of discontinuity

x c

"Removing" the Discontinuity

- A discontinuity at c is called removable if
- If the function can be made continuous by
- defining the function at x c
- or redefining the function at x c
- Go back to this link, determine which (if any) of

the discontinuities can be removed

Which of These is Dis/Continuous?

- When x 1 why or not

Are any removable?

Continuity Theorem

- A function will be continuous at any number x

c for which f(c) is defined, when - f(x) is a polynomial
- f(x) is a power function
- f(x) is a rational function
- f(x) is a trigonometric function
- f(x) is an inverse trigonometric function

Properties of Continuous Functions

- If f and g are functions, continuous at x

cThen - is continuous (where s is a

constant) - f(x) g(x) is continuous
- is continuous
- is continuous
- f(g(x)) is continuous

One Sided Continuity

- A function is continuous from the right at a

point x a if and only if - A function is continuous from the left at a point

x b if and only if

a

b

Continuity on an Interval

- The function f is said to be continuous on an

open interval (a, b) if - It is continuous at each number/point of the

interval - It is said to be continuous on a closed interval

a, b if - It is continuous at each number/point of the

interval and - It is continuous from the right at a and

continuous from the left at b

Continuity on an Interval

- On what intervals are the following functions

continuous?

Intermediate Value Theorem

- Given function f(x)
- Continuous on closed interval a, b
- And L is a number strictly between f(a) and f(b)
- Then there exists at least one number c on the

open interval (a, b) such that f(c) L

f(b)

L

f(a)

b

a

Locating Roots with Intermediate Value Theorem

- Given f(a) and f(b) have opposite sign
- One negative, the other positive
- Then there must be a root between a and b

a

Try exercises 88, 90, and 92 pg 100

b

Assignment

- Lesson 2.4A
- Page 98
- Exercises 1 49 odd
- Lesson 2.4B
- Page 99
- Exercises 51 75, 85 105 odd

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