Math 151 Section 2.5
Definition:
Continuity
A function, f ( x ), is continuous at a if and only if lim f ( x )  f ( a ).
xa
A function is discontinuous (not continuous) at
i)
a if any of the following occur:
f ( a ) is undefined
lim f ( x ) exists but lim f ( x )  f ( a )
ii)
xa
iii)
xa
lim f ( x ) does not exist
xa
If
f (x ) is continuous at every point where it is defined then it is continuous on its domain.
If
lim f ( x ) exists but lim f ( x )  f ( a ) , the discontinuity is called removable. In this case, we
xa
xa
only need to change the definition of the function at one point to remove the discontinuity.
Left and right continuity:
A function, f ( x ), is right continuous at a if and only if lim  f ( x )  f ( a ).
xa
A function, f ( x ), is left continuous at a if and only if lim  f ( x )  f ( a ).
xa
The Intermediate Value Theorem: If
f (x ) is continuous on [ a , b ] and N is any number between
f (a ) and f (b ) then then there is a c in [ a , b ] for which f ( c )  N .
f ( x)  x 2  2 x  5
Example: Show that
has a root in the interval [1, 2 ] .
Examples: Find the discontinuities of each function. Give a reason for each. Is the discontinuity
removable and how should it be removed? Is the function continuous on its domain? Is the function left
or right continuous or neither?
1. a)
2.
f ( x) 
x2  2x
x  5x
2
3 x 2  2 x x  1
f ( x)  
1 x
 x4
b)
 x2  2x

f ( x)   x 2  5 x

 1
x  0, x  5
x  0, x  5
3.
4.
5.
 1
sin  
f ( x)    x 

 0

1
 x sin  
f ( x)  
x

0

cx 2  c
f ( x)  
 2x  c
x0
x0
x0
x0
x2
2 x
Find the value of
c that makes the function continuous at 2.