Abc

1. 2.3 Continuity
Grand Canyon, Arizona
Greg Kelly, Hanford High School, Richland, Washington
Photo by Vickie Kelly, 2002

2. Most of the techniques of calculus require that functions
be continuous. A function is continuous if you can draw it
in one motion without picking up your pencil.

A function is continuous at a point if the limit is the same
as the value of the function.
This function has discontinuities
at x=1 and x=2.
It is continuous at x=0 and x=4,
because the one-sided limits
match the value of the function
1 2 3 4
1
2

3. jump infinite oscillating
Essential Discontinuities:
Removable Discontinuities:
(You can fill the hole.)


4. Removing a discontinuity:
 
3
2
1
1
x
f x
x



has a discontinuity at .
1
x 
Write an extended function that is continuous
at .
1
x 
3
2
1
1
lim
1
x
x
x



  
  
2
1
1 1
lim
1 1
x
x x x
x x

  

 
1 1 1
2
 

3
2

 
3
2
1
, 1
1
3
, 1
2
x
x
x
f x
x
 


 
 
 


Note: There is another
discontinuity at that can
not be removed.
1
x  

5. Removing a discontinuity:
 
3
2
1
, 1
1
3
, 1
2
x
x
x
f x
x
 


 
 
 


Note: There is another
discontinuity at that can
not be removed.
1
x  
-5
-4
-3
-2
-1
0
1
2
3
4
5
-5 -4 -3 -2 -1 1 2 3 4 5


6. Continuous functions can be added, subtracted, multiplied,
divided and multiplied by a constant, and the new function
remains continuous.
Also: Composites of continuous functions are continuous.
examples:  
2
sin
y x
 cos
y x



7. Intermediate Value Theorem
If a function is continuous between a and b, then it takes
on every value between and .
 
f a  
f b
a b
 
f a
 
f b
Because the function is
continuous, it must take on
every y value between
and .
 
f a
 
f b


8. Example 5: Is any real number exactly one less than its cube?
(Note that this doesn’t ask what the number is, only if it exists.)
3
1
x x
 
3
0 1
x x
  
  3
1
f x x x
  
 
1 1
f    
2 5
f 
Since f is a continuous function, by the
intermediate value theorem it must
take on every value between -1 and 5.
Therefore there must be at least one
solution between 1 and 2.

Use your calculator to find an approximate solution.
 
3
solve 1,
x x x
 
F2 1: solve
1.32472

9. This example was graphed
on the classic TI-89. You
can not change the
resolution on the Titanium
Edition.
Graphing calculators can sometimes make non-
continuous functions appear continuous.
Graph:  
floor
y x

CATALOG F floor(
Note resolution.
The calculator “connects the dots”
which covers up the discontinuities.

10. Graphing calculators can make non-continuous
functions appear continuous.
Graph:  
floor
y x

CATALOG F floor(
GRAPH
The open and closed circles do not
show, but we can see the
discontinuities. p
If we change the plot style
to “dot” and the resolution
to 1, then we get a graph
that is closer to the
correct floor graph.