Page 1 | © 2012 by Janice L. Epstein 2.5 Continuity Page 2 | © 2012 by Janice L. Epstein 2.5 Continuity Continuity (Section 2.5)
EXAMPLE 2
Explain why each function is discontinuous at the given point.
x 2 -1
a) a = -1, f ( x) =
x +1
f ( x) is continuous at a number a if lim f ( x) = f (a )
x a
5
f ( x)
EXAMPLE 1
Where is the graph discontinuous?
-5
5
x
f ( x) is continuous from the right at a number a if
lim+ f ( x) = f (a )
x a
f ( x) is continuous from the left at a number a if
lim- f ( x) = f (a )
x a
b) a = 4 ,
ìï x 2 - 2 x - 8
ï
if x ¹ 4
f ( x) = ïí x - 4
ïï
if x = 4
ïî3
A function f ( x) is continuous on an interval if it is continuous at
every number in the interval. At the endpoints it is understood that
the function is left or right continuous.
Continuity Rules
If f and g are continuous at a and c is a constant, then the following
functions are also continuous at a
f
f g
cf
fg
(provided that g (a) ¹ 0 )
g
A polynomial is continuous on  = (-¥, ¥)
A rational function is continuous on its domain
If n is a positive even integer, f ( x ) = n x is continuous on [0, ¥)
If n is a positive odd integer, f ( x) is continuous on (-¥, ¥)
Page 3 | © 2012 by Janice L. Epstein 2.5 Continuity Page 4 | © 2012 by Janice L. Epstein 2.5 Continuity EXAMPLE 3
Find where the given functions are discontinuous
ì
2 x + 1 if x £ -1
ï
ï
ï
a) g ( x ) = í3 x
if -1 < x < 1
ï
ï
ï
ï
î2 x + 1 if x ³ 1
EXAMPLE 4
Find the value or values of c that make f continuous on (-¥, ¥)
ïìï x 2 - c 2 if x < 4
f ( x) = í
ïïîcx + 20 if x ³ 4
If g is continuous at a and f is continuous at g (a ) then
( f  g )( x) = f ( g ( x)) is continuous at a.
b) h(t ) =
t +1
4t 2 -1
ì
ï -x
ï
ï
c) h( x) = ï
í1
ï
ï
ï
ï
î x
EXAMPLE 5
Find where the given functions are discontinuous
a) F ( x) = 2 x + 25 - x 2
if x £ 0
if 0 < x < 1
b) g (t ) =
1
t + t2 -4
if x ³ 1
c) G (t ) = x - x 2
d) H ( x) =
x-2
5+ x
Page 5 | © 2012 by Janice L. Epstein 2.5 Continuity Page 6 | © 2012 by Janice L. Epstein 2.5 Continuity A removable discontinuity at a is a discontinuity that can be
removed by redefining the function at a.
Intermediate Value Theorem
Suppose f is continuous on the closed interval [ a, b ] and let N be
any number strictly between f (a) and f (b) . Then there exists a
number c in (a, b) such that f (c) = N .
EXAMPLE 6
Which of the following functions f has a removable discontinuity
at a? If it is removable, redefine f so that the function is
continuous on  .
a) a = -2,
b) a = 7,
c) a = -4,
f ( x) =
f ( x) =
x2 - 2 x - 8
x+2
x-7
x-7
f ( x) =
EXAMPLE 7
Use the Intermediate Value Theorem to show there is a root of the
given equation in the given interval.
a) x 5 - 2 x 4 - x - 3 = 0, (2,3)
b) x 2 = x + 1, (1, 2)
x3 + 64
x+4
EXAMPLE 8
Use the Intermediate Value Theorem to show that there is a
positive number c such that c 2 = 2 .
d) a = 9,
f ( x) =
3- x
9- x