Warm-Up/ActivatorSketch a graph you would describe as continuous.

Sketch a graph you would describe as discontinuous.

ContinuityEssential Question:What are the characteristics of a continuous function?

Continuity

Where am I continuous?

Where am I discontinuous?

xx

xx

xx

xx

xx

xf

3,10

30,23

02,

25,4)4(

5,12

)( 2

3

Definition of ContinuityLet c be a number in the interval (a,b) and let f be a function whose domain contains the interval (a,b). The function f is continuous at the point c if the following conditions are true.

1. f(c) is defined

2. exists

3. )()(lim cfxfcx

)(lim xfcx

Continuous IntervalsIf f is continuous at every point in the interval

(a,b) then it is continuous on the interval (a,b)

The domain of the function determines continuity.A polynomial function is continuous at every

real number.A rational function is continuous at every

number in its domain.

Example 1]

f(x) = x2 - 2x + 3 f(x) = x3 - x

Example 2Finding DiscontinuitiesDetermining Continuity of a Function  A. f(x) =

B. f(x) =

C. f(x) =

x

1

1

12

x

x

1

12 x

Removable vs Non-removable

107

103)(

2

2

xx

xxxf

Holes are removable

Vertical asymptotes (Infinite Discontinuities) and jump discontinuities are non-removable.

Continuity on a Closed Interval

Let f be defined on a closed interval [a,b]. If f is continuous on the open interval (a,b) and

and

then f is continuous on the closed interval [a,b]. Moreover, f is continuous from the right at a and continuous from the left at b.

)()(lim afxfax

)()(lim bfxfbx

Examining Continuity at Endpoints

xxf 3)(

Examining Continuity at Endpoints

32,1

21,5)(

2 xx

xxxf

Greatest Integer FunctionThe Greatest Integer Function - is a step

function

or [[x]] = greatest integer less than or equal to x

x

Modeling a Cost FunctionA bookbinding company produces 10,000 books in

an 8-hour shift. The fixed costs per shift amount to $5000, and the unit cost per book is $3. Using the greatest integer function, you can write the cost of producing x books as

xx

C 3]])10000

1[[1(5000

Sketch the graph of this cost function

Cost Function Graph

Compound InterestBanks and other financial institutions differ on how

interest is paid to an account. If the interest is added to the account so that future interest is paid on previously earned interest, then the interest is said to be compounded. Suppose, for example, that you deposit $10,000 in an account that pays 6% interest, compounded quarterly. Because the 6% is the annual interest rate, the quarterly rate is 1/4(.06) = 0.015 or 1.5%.

Compound InterestSketch the graph of the balance in the

account described above. A = 10000(1+0.015)^[[4t]]