Real Analysis – Limits and Continuity
— 6. Limits and Continuity —
After introducing sequences and gaining some knowledge of some of their properties (I, II, III, and IV) we are ready to embark on the study of real analysis.
— 6.1. Preliminary Definitions —
Physics is expressed best and most powerfully in the language of mathematics and a very useful mathematical concept for physics is the concept of a function.
Generally speaking a function is an association (it transforms an input signal from the first set into an output signal of a second set) between the elements of two sets.
The sequences we studied are a special case of functions: they take natural numbers (or a subset of them) as their input signals and map them to real numbers.
Now, more formally we introduce:
Sometimes we may not be interested in how the function maps the whole of but just on a particular subset of
. So it makes sense to introduce:
Definition 24
Given |
As we did for sequences we can too define what is a bounded above function, a bounded below function, a bounded function and etc.
As an example we’ll give:
Definition 25 |
— 6.2. Introduction to Topology —
We will now introduce some light topological notions in order to shed some light into the study of limits and continuity.
Once again so that we don’t let things get too abstract let us give an example:
It is easy to see (and we won’t give a rigorous proof of that) that and that
is the only isolated point of
.
Definition 28 The symbol |
As an example let us calculate
In this case it is and
so that the limit we intend to calculate indeed makes sense.
If is a sequence of points in
such as
then it follows that
As an application of theorem 28 let us calculate the following limit
It is easy to see that this limit doesn’t exist. Let it is
and
.
Since the limit from the left is different from the limit from the right we can conclude that doesn’t exist.
Definition 29
|
If you’re having trouble understanding these definitions just think that if isn’t bounded above than it means that
.
Which is just the definition of limit point.
Definition 30
|
Definition 31 Let
|
We’ll only define the limit of a function in limit points of the domain. Notice that by this way we can too define the limit of points that don’t belong in the domain of the function.
As always a few examples will be provided in order for us to test our knowledge.
- Calculate the limit of
.
and
since
isn’t bounded above in
. Thus the limit we set ourselves to calculate makes sense in our theory of limits.
Let
be a sequence of points in
such as
and
, then
and it always is
.
- Calculate the limit of
Choosing
we see that the domain is
. Thus
Let us choose
. Thus
and
.
In this case it trivially is
.
Now if we choose
it also is
but
and so
.
Thus we were able to find
,
such as
but
. Thus
doesn’t exist.
In order for us to proceed deeper in the study of limits and continuity we’ll introduce the notions of one-sided limit. We’ll use the symbols to denote approximation to
by real numbers that are bigger than
. In an analogous way we can also define
to denote the approximation to
by real numbers that are smaller than
.
Formalizing the previous notions:
Definition 32
|
December 2, 2010 at 5:08 am
How to type Latex theorem environment as above? please tell me!
Thanks!
December 2, 2010 at 10:17 am
I just sent you an email with the modification I made to the latex2wp script.
Let me know if you found it useful.
May 26, 2012 at 12:45 pm
Thanks for let me know latex2wp.
But the formula in my post is showed text like
WP Latex plugin is installed as but it becomes “formula is not prase”.
How you make it ?