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Subject: Continuity and Differentiability
Date: Fri, 13 Nov 2009 14:26:49 -0700
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Continuity and Differentiability
Continuity & Differentiability=20
miscellaneous on-line topics for
Calculus Applied to the Real World =
Part A:=20
Continuity=20
|
|
Return to Main Page
Part B:=20
Differentiability
Exercises for=20
This Topic
=
Index of=20
On-Line Topics
Everything for=20
Calculus
=
Everything for=20
Finite Math
<=
FONT=20
face=3D"arial, helvetica, sans-serif" color=3Dblue =
size=3D2>Everything for=20
Finite Math & Calculus
Utility:=20
Function Evaluator & Grapher=20
|
Part A: Continuity=20
Note To understand this topic, you will need to be familiar =
with=20
limits, as discussed in the chapter on derivatives in Calculus Applied to the Real World. If you =
like, you=20
can review the topic=20
summary material on limits or, for a more detailed study, the on-line=20
tutorial on limits.=20
To begin, we recall the definition (see Section 6 in the derivatives =
chapter=20
in Calculus Applied to the Real =
World.) of=20
what it means for a function to be continuous at a point or on a subset =
of its=20
domain.=20
Continuous at a Point =
The=20
function f is continuous at the point a in its domain if:=20
| 1. |
lim
xa |
f(x) exists; |
| 2. |
lim
xa |
f(x) =3D =
f(a) | If f is not=20
continuous at a, we say that f is discontinuous at a.=20
Note
If the point a is =
not in the=20
domain of f, we do not talk about whether or not f is continuous =
at a.=20
Continuous on a Subset of the =
Domain=20
The function f is continuous on the subset S of its =
domain if=20
it continuous at each point of S.=20
Examples
1. All =
closed=20
form functions are continuous on their (whole) domain. A =
closed-form=20
function is any function that can be obtained by combining =
constants,=20
powers of x, exponential functions, radicals, logarithms, and=20
trigonometric functions (and some other functions we shall not =
encounter=20
in this text) into a single mathematical formula by means =
of the=20
usual arithmetic operations and composition of functions. Examples =
of=20
closed form functions are:=20
| 3x2 - x + 1, =
=20
|
2x
![](3D"http://people.hofstra.edu/Stefan_Waner/realworld/calctopic1/SYMB/F=)
x + 3 |
, and |
ex2-1. |
2.=20
The function f(x) =3D 1/x is continuous at every point of its =
domain.=20
(Note that 0 is not a point of the domain of f, so we don't =
discuss what=20
it might mean to be continous or discontinuous there.)=20
3. The function=20
| f(x) =3D |
![](3D"http://people.hofstra.edu/Stefan_Waner/realworld/calctopic1/SYMB/L=) |
-1
x2+x |
|
if x =20
2
if x > =
2 | is not a=20
closed-form function (since we need two algebraic formulas to =
specify it).=20
Moreover, it is not continuous at x =3D 2, since limx2f(x)=20
does not exist. |
Example 1 Recognizing Points of =
Discontinuity in=20
a Graph
Let f have the graph shown below.
At which points of the domain of f is the function f =
discontinuous?
Solution
According to the definition, f can fail to be continuous at a point a =
of its=20
domain if either:=20
| 1. |
lim
xa |
f(x) does not exist, or |
| 2. |
lim
xa |
f(x) exists, but =20
f(a) |
Looking at the figure, we see that the possible points where things =
might go=20
wrong are at x =3D -1, 0, 1, and 2. Let us =
look at these=20
points one-at-a-time.=20
x =3D -1: Notice that x =3D -1 is a point of the domain of f, and lim =
x-1f(x) does not exist, because the left- and =
right-habd limits disagree. Therefore, f has a discontinuity at x =3D =
-1. =20
x =3D 0: From the graph, we see that f(0) is not defined. =
Therefore, 0=20
is not in the domain of f, so we cannot say that f has a discontinuity =
at x =3D 0.=20
(Some authors would say that f is discontinuous at x =3D 0, but we don't =
consider=20
such points...)=20
x =3D 1: At the point where x =3D 1, the limit does exist, =
however,=20
lim
x1 |
f(x) =3D 1, but |
| f(1) |
=3D 2 |
Thus, since the =
limit at 1 does not=20
equal the value of the function, we have a discontinuity at x =3D 1.=20
x =3D 2: Although there is a cusp in the graph at x =3D 2, you =
will see=20
that the limit at x =3D 2 exists, and agrees with f(2). Therefore, f is =
continuous=20
at x =3D 2.=20
Thus, the only points of discontinuity in the domain of f occur at x =
=3D -1 and x =3D1.=20
Here is a more interactive example.=20
Example 2 Recognizing Points of =
Discontinuity in=20
a Graph
Let f have the graph shown below.
The next examples discuss functions specified algebraically rather =
than=20
geometrically.=20
Example 3 Identifying Points of =
Discontinuity in=20
a Non-Closed Form Function
Let f be specified as=20
| f(x) =3D |
|
x2+4
x + 1
x2 + 1 |
|
if x =20
0
if 0 < x =20
1
if x > 1 |
. |
Then, the only conceivable =
points of=20
discontinuity occur at x =3D 0 and x =3D 1. (At all other points in the =
domain, f=20
can be specified by a single closed-form formula; for example, near x =
=3D 0.5, f=20
can be specified as x + 1.)=20
Looking near x =3D 0, we get=20
lim
x0- |
f(x) =3D 4 |
Substitute x =3D 0 in the first (closed-form)=20
formula |
lim
x0+ |
f(x) =3D 1 |
Substitute x =3D 0 in the second (closed-form)=20
formula |
Since these limits disagree, limx0f(x)=20
does not exist, and so the function is discontinuous at x =3D 0.=20
If you now compute the left- and right limits at x =3D 1, on the =
other hand,=20
you will fined they agree and equal 2, which is also the value of f(1). =
Thus, f=20
is continuous at x =3D 1, and so the only point of discontinuity is x =
=3D 0.=20
Example 4 Fixing up a Function =
Let f be given by=20
| f(x) =3D |
|
x2 - x + =
1
kx2 +=20
1 |
|
if x =20
1
if x > 1 |
. |
For which value(s) of k is f =
continuous on=20
its domain?=20
Solution
Since the only place where something can go wrong is at x =3D 1, we =
look at the=20
left- and right limits, as well as the value of the function.=20
lim
x1- |
f(x) =3D 1 |
lim
x1+ |
f(x) =3D k+1 |
| f(1) =3D 1 |
To make =
the left- and right=20
limits agree, we must make k =3D 0 (and this also makes f(1) agree with =
those=20
limits). Thus, we take k =3D 0.=20
You can now either go on and try those exercises that deal with =
continuity in=20
the exercise=20
set for this topic, or first finish the on-line text by going to Part=20
B: Differentiability.
Return to Main Page
Part B:=20
Differntiability
Exercises for=20
This Topic
=
Index of=20
On-Line Topics
Everything for=20
Calculus
=
Everything for=20
Finite Math
<=
FONT=20
face=3D"arial, helvetica, sans-serif" color=3Dblue =
size=3D2>Everything for=20
Finite Math & Calculus
Utility:=20
Function Evaluator & Grapher=20
|
Last Updated:October, 1999
Copyright =A9 1999 StefanWaner and Steven R. Costenoble=20
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