1.4 Continuity and One-Sided Limits - PowerPoint Presentation

Slide1

1.4 Continuity and One-Sided LimitsObjective: Determine continuity at a point and on an open interval; determine one-sided limits and continuity on a closed interval.

Miss

Battaglia

AB/BC CalculusSlide2

What does it mean to be continuous?

Continuity

Below are three values of x at which the graph of f is NOT continuous

At all other points in the interval (

a,b), the graph of f is uninterrupted and continuous

f(c) is not defined

d

oes not existSlide3

Definition of Continuity

Continuity at a Point:

A function f is continuous at c if the following three conditions are met.

1. f(c) is defined

2. exists 3.Continuity on an Open Interval: A function is continuous on an open interval (a,b

) if it is continuous at each point in the interval. A function that is continuous on the entire real line (-∞,∞) is everywhere continuous.Slide4

Removable (f can be made continuous by appropriately defining f(c)) & nonremovable.

Discontinuities

Removable Discontinuity

Nonremovable

DiscontinuitySlide5

Continuity of a Function

Discuss the continuity of each functionSlide6

One-Sided Limits and Continuity on a Closed Interval

Limit from the right

Limit from the left

One-sided limits are useful in taking limits of functions involving radicals (Ex: if n is an even integer)Slide7

A One-Sided LimitFind the limit of as x approaches -2 from the right.Slide8

One sided limits can be used to investigate the behavior of step functions

. A common type is the

greatest integer function

defined by

= greatest integer n such that n < xEx: = 2 and

= -3Find the limit of the greatest integer function as x approaches 0 from the left and from the right.

 The Greatest Integer Function

 Slide9

Definition of Continuity on a Closed IntervalA function f is continuous on the closed interval [

a,b

] if it is continuous on the open interval (

a,b

) and andThe function f is continuous from the right at a and continuous from the left at b.

Theorem 1.10: The Existence of a Limit

Let f be a function and let c and L be real numbers. The limit of f(x) as x approaches c is L iffand Slide10

Continuity on a Closed IntervalDiscuss the continuity ofSlide11

Theorem 1.11 PROPERTIES OF CONTINUITY

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

Scalar multiple:

bf

Sum or difference:

f + gProduct: fgQuotient:

, if g(c)≠0 

By Thm 1.11, it follows that each of the following functions is continuous at every point in its domain.Slide12

THEOREM 1.12 CONTINUITY OF A COMPOSITE FUNCTION

If g is continuous at c and f is continuous at g(c), then the composite function given by

is continuous at c.

Theorem 1.13 INTERMEDIATE VALUE THEOREM

If f is continuous on the closed interval [

a,b

],

and k is any number between f(a) and f(b), then there is at least one number in c in [

a,b] such that f(c) = k

 Slide13

Consider a person’s height. Suppose a girl is 5ft tall on her thirteenth bday and 5ft 7in tall on her fourteenth

bday

. For any height, h, between 5ft and 5ft 7in, there must have been a time, t, when her height was exactly h.

The IVT guarantees the existence of

at least one number c in the closed interval [a,b]Intermediate Value TheoremSlide14

An Application of the IVT

Use the IVT to show that the polynomial function f(x)=x

3

+ 2x – 1 has a zero in the interval [0,1]Slide15

AB: Page 78 #27-30, 35-51 odd, 69-75 odd, 78, 79, 83, 91, 99-102BC: Page 78 #3-25 every other odd, 31, 33, 34, 35-51 every other odd, 61, 63, 69, 78, 91,99-103

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