We now want to combine some of the concepts that we have introduced before:
functions, sequences, and topology. In particular, if we have some function
f(x) and a given sequence { an }, then we can apply the
function to each element of the sequence, resulting in a new sequence. What we
would want is that if the original sequence converges to some number L,
then the new sequence { f( an )} should converge to f(L),
and if the original sequence diverges, the new one should diverge also. This
seems not too much to ask for, but is quite simple minded.
Example:
-
Consider the function f, where f(x) = 1 if x
0 and f(x) = 2 if x > 0.
- The sequence { 1/n } converges to 0. What happens to the
sequence { f( 1/n ) } ?
- The sequence { 3 + (-1)n } is divergent. What
happens to the sequence { f ( 3 + (-1)n ) } ?
- The sequence { (-1)n / n } converges to zero. What
happens to the sequence { f ( (-1)n / n ) } ?
As the above easy example shows, things can be more complicated than
anticipated. Therefore, we have to attack the problem more systematically.
First, we need to define what we mean by 'limit of a function'.
- A function f with domain D in R converges to a limit L
as x approaches a number c if for any sequence { xn
in D that converges to c the sequence { f ( xn )
} converges to L.
We write
f(x) = L
-
Apply this definition in these cases:
- Let f(x) = m x + b. Then does the limit of that function
exist at an arbitrary point x ?
- Let g(x) = [x], where [x] denotes the greatest integer
less than or equal to x. Then does the limit of g exist at
an integer ? How about at numbers that are not integers ?
- In the above definition, does c have to be in the domain D
of the function ? Is c in the closure(D) ? Do you know a
name for c in terms of topology ?
The above definition works quite well to show that a functionis not continuous,
because you only have to find one particular sequence whose images do not
converge as a sequence. It is not a good definition, in general, to prove
convergence of a function, because you will have to check every possible
convergent sequence, and that is hard to do. We would therefore like another
definition of convergence or limit of a function.
- A function f with domain D in R converges to a limit
L as x approaches a number c
closure (D) if: given any
> 0 there exists a
> 0 such that:
- if x
D and | x - c | <
then | f(x) - L | <
-
Consider the function f with f(x) = 1 if x is rational
and f(x) = 0 if x is irrational. Does the limit of f(x)
exist at an arbitrary number x ?
Regardless of which of the two definitions might be considered easier to use in
a particular situation, the basic problem right now is that we have two
different definitions for the same concept. We therefore have to show that both
definitions are actually equivalent to each other.
- If f is any function with domain Din R, and c
closure(D) then the following are equivalent:
- For any sequence { xn } in D that converges
to c the sequence { f ( xn ) } converges to
L
- given any
> 0 there exists a
> 0 such if x
closure(D) and | x - c | <
then | f(x) - L | <
In other words, both definitions of continuity are equivalent, and we can use
which ever seems the easiest. Here are some basic properties of limits of
functions.
- If
f(x) exists, the limit is unique.
-
[
f(x) + g(x) ] =
f(x) +
g(x), provided that
f(x) and
g(x)
exist.
-
[ f(x) g(x) ] =
f(x)
g(x), provided that
f(x) and
g(x) exist.
-
[ f(x) / g(x) ] =
f(x) /
g(x), provided that
f(x) and
g(x) exist and
g(x) # 0.
Sometimes a function may not have a limit using the above definitions, but
when the domain of the function is restricted, then a limit exists. This leads
to the concepts of one-sided limits.
- If f is a function with domain D and c
closure(D). Then:
- f has a left-hand limit L at c if for
every
> 0 there exists
> 0 such that if x
D and c -
< x < c then | f(x) - L | <
.
We write
f(x) = L.
- f has a right-hand limit L at c if for
every
> 0 there exists
> 0 such that if x
D and c < x < c +
then | f(x) - L | <
.
We write
f(x) = L.
This is the formal definition of x approaching c either only from
the right side, or only from the left side. These one-sided limits are related
to regular limits in a straight forward manner:
-
f(x) = L if and only if
f(x) = L and
f(x) = L
Now that we have some idea about limits of functions, we will move to the
next question: if some sequence converges to c, and the function
converges to L as x approaches c, then when is it true that
f(c) = L ? This will be the contents of the next section, continuity.
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