Warm-Up/Activator
 Sketch a graph you would describe as continuous.
 Sketch a graph you would describe as discontinuous.
Essential Question:
What are the characteristics of a
continuous function?
Continuity
 Where am I continuous?
 Where am I discontinuous?
x  5
  x  12,
( x  4) 3  4,  5  x  2

f ( x)  
x2 ,
2 x0
  3 x  2,
0 x3

 x  10,
3 x
Definition
of
Continuity
 Let 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.
lim f ( x) exists
x c
f ( x )  f (c )
3. lim
x c
Continuous Intervals
 If 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 2
Finding Discontinuities
 Determining Continuity of a Function
A. f(x) = 1
x
x 2 1
B. f(x) =
x 1
1
C. f(x) = 2
x 1
Removable vs Non-removable
x  3x  10
f ( x)  2
x  7 x  10
2
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
lim f ( x)  f (a ) and
xa
lim f ( x)  f (b)
x b
 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.
Examining Continuity at Endpoints
f ( x)  3  x
Examining Continuity at Endpoints
5  x,1  x  2
f ( x)   2
 x  1,2  x  3
Greatest Integer Function
 The Greatest Integer Function - is a step function
x or [[x]] = greatest integer less than or equal to x
Modeling a Cost Function
 A 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
x 1
C  5000(1  [[
]])  3x
10000
Sketch the graph of this cost function
Cost Function Graph
Compound Interest
 Banks 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 Interest
 Sketch the graph of the balance in the account
described above. A = 10000(1+0.015)^[[4t]]