This document discusses continuity of functions and discontinuities. It defines a continuous function as one where the limit matches the value of the function at a point. There are two types of discontinuities: essential discontinuities which cannot be removed and removable discontinuities which can be removed by defining the function at the point of discontinuity. The intermediate value theorem states that if a function is continuous on an interval, it must take on all values between the function values at the endpoints.

1. 2.3 Continuity
Grand Canyon, Arizona
Greg Kelly, Hanford High School, Richland, WashingtonPhoto 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.)
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4. Removing a discontinuity:
( )
3
2
1
1
x
f x
x
−
=
−
has a discontinuity at .1x =
Write an extended function that is continuous
at .1x =
3
21
1
lim
1x
x
x→
−
−
( ) ( )
( ) ( )
2
1
1 1lim
1 1x
x x x
x x→
− + +=
+ −
1 1 1
2
+ +
=
3
2
=
( )
3
2
1
, 1
1
3
, 1
2
x
x
xf x
x
 −
≠ −= 
 =

Note: There is another
discontinuity at that can
not be removed.
1x = −

5. Removing a discontinuity:
( )
3
2
1
, 1
1
3
, 1
2
x
x
xf x
x
 −
≠ −= 
 =

Note: There is another
discontinuity at that can
not be removed.
1x = −
→

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
siny x= cosy x=
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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
1x x= −
3
0 1x x= − −
( ) 3
1f x x x= − −
( )1 1f = − ( )2 5f =
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.
( )3
solve 1,x x x= −
1.32472
Use your calculator to find an approximate solution.
menu 3
Algebra
1
Solve
→

9. Now press to ask the calculator for an
approximate answer, and notice how much quicker it is.
enterctrl
( )3
solve 1,x x x= −
1.32472
We will find the same command another way.
π
Functions, commands and tools on the calculator can often
be found in more than one way.
If you can’t find something, look in the catalog.
S solve(
If you press it takes the calculator a while to find
the answer, because it is trying to find an exact answer first.
enter
Use the up arrow to highlight the formula, and press
to make another copy.
enter

10. Now press to ask the calculator for an
approximate answer, and notice how much quicker it is.
enterctrl
( )3
solve 1,x x x= −
1.32472
We will find the same command another way.
π
Functions, commands and tools on the calculator can often
be found in more than one way.
If you can’t find something, look in the catalog.
S solve(
If you press it takes the calculator a while to find
the answer, because it is trying to find an exact answer first.
enter
Use the up arrow to highlight the formula, and press
to make another copy.
enter