On-line Math 21

On-line Math 21

1.1  Continuity

A function f(x) is continuous at a point a if it behaves as it should there, that is:

Definition 1 Assume that f(x) is defined in some open interval that contains a . Then, f is continuous at a if:

lim
x® a 
 f(x) = f(a).

f is continuous on the interval (b,c) if it is continuous at all points a Î (b,c) .

Note that this says two things, that the limit exists, and that the function's value at a is the same as the limit.

The usual idea of continuity is that a graph is continuous if you can draw the picture without lifting your pencil from the paper. That's a little limited, but gives the idea well enough.

Figure
There is also a definition of one-sided continuity (continuous with respect to limits from one side or the other). This concept is primarily useful only to discuss whether or not the function is continuous at the endpoints of its domain. If f is defined on [a,b] it will be continuous from the right at a if if:

lim
x¯ a 
 f(x) = f(a).

A similar definition can be made for continuity from the left.

Examples

Example 2 Let
f(x) = ì
í
î
3x+1,
if x ³ 1
x+2,
if x < 1
 .

Figure
Then f is not continuous at a = 1 . This is because

lim
x® 1 
 f(x)
does not exist, which you should recall from an earlier section.

Example 3 Let
f(x) = ì
í
î
x2,
if x ¹ 1
5,
if x = 1
 .
This function is also not continuous at x = 1 , since

lim
x® 1 
 f(x) = 1,
even though f(1) = 5 . The difference here is that the function could have been continuous, if it weren't for the fact that it was ``poorly'' defined at x = 1 .

Example 4 Let
f(x) = ì
í
î
3x+1,
if x ³ 1
x+3,
if x < 1
 .
Then f is continuous at a = 1 . This is because

lim
x® 1 
 f(x) = 4
does exist, and is the same as f(1) .

Figure
Example 5 Since, for any polynomial f(x) = anxn+... +a1x+a0 , and for any x = a ,

lim
x® a 
 f(x) = f(a),
any polynomial is continuous. We say that a function f is continuous, instead of continuous at a point or on an interval, if it is continuous at every point of its domain.

Example 6 f(x) = |x| is continuous, even though it has ``bad behavior'' at x = 0 . Its behavior is just not that bad there. The limit of f(x) , as x approaches 0, is 0, and since that is the value of the function, then it is continuous there. At other points, f is just like a polynomial in some open interval (either x or -x ), so it is continuous.

Exercises

Exercise 2 Find the values of a for which the function
f(x): = ì
í
î
x2-2x,
if x £ 2
ax+4,
if x > 2,
will be continuous at x = 2 . Answer: a = .

Exercise 3 Is there a way to define the value of f(0) so that the function
f(x) = x
|x|
will be continuous at x = 0 ? Answer: f(0) = .

Copyright (c) 2000 by David L. Johnson.

File translated from TEX by TTH, version 2.61.
On 18 Oct 2000, 00:16.