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Continuity
Grand Canyon, ArizonaGreg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2002
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
jump infinite oscillating
Essential Discontinuities:
Removable Discontinuities:
(You can fill the hole.)
Removing a discontinuity:
3
2
1
1
xf x
x
has a discontinuity at .1x
Write an extended function that is continuous at .1x
3
21
1lim
1x
x
x
2
1
1 1lim 1 1x
x x xx x
1 1 1
2
3
2
3
2
1, 1
13
, 12
xx
xf x
x
Note: There is another discontinuity at that can not be removed.
1x
Removing a discontinuity:
3
2
1, 1
13
, 12
xx
xf x
x
Note: There is another discontinuity at that can not be removed.
1x
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: 2siny x cosy x
Definition of Continuity
Continuity at a point: A function f is continuous at c if the following three conditions are met
1. f c is defined
2. limx cf x exists
3. limx cf x f c
A function is continuous on an open interval
If it is continuous at each point in the interval. A
function that is continuous on the entire real line is everywhere
continuous
,a b
Formal Definition of Continuityon an Interval
A function f is continuous on a closed interval
,a b
,a b
If it is continuous on the open interval and
limx a
f x f a
and limx b
f x f b
The function f is continuous from the right at
and continuous from the left at
ab
Intermediate Value Theorem (IVT)
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
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
30 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.
Use your calculator to find an approximate solution.
3
1
2 1
Y x
Y x
or 31 1Y x x
Graphing calculators can make non-continuous functions appear continuous.
The calculator “connects the dots” which covers up the discontinuities.
Use your calculator to graph the function
1
2
xf x
x
Graphing calculators can make non-continuous functions appear continuous.
Graph: intf x x
GREATEST INTEGER FUNCTION
1 2 3 4 5 6 7-1-2-3-4-5-6-7
1
2
3
4
5
-1
-2
-3
-4
-5
x
y
Graphing calculators can make non-continuous functions appear continuous.
If we change the plot style to “dot”, we get a graph that is closer to the correct graph of the function.
The open and closed circles do not show, but we can see the discontinuities.
1 2 3 4 5 6 7-1-2-3-4-5-6-7
1
2
3
4
5
-1
-2
-3
-4
-5
x
y