Real Analysis – Limits and Continuity
— 6. Limits and Continuity —
— 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:
Given it is is the image of by .
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 is said to be bounded iff|
— 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 27 We’ll use the symbol to denote approximation to by real numbers bigger than .
In an analogous way we can also define .
Thus, we define if for all such as corresponds a sequence such as .
|Definition 28 The symbol will be used to denote and the symbol will denote|
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 is a limit point of if isn’t bounded above in .
is a limit point of if isn’t bounded below in .
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 is said to be a limit point of if
|Definition 31 Let , , and .
has limit in point if for all sequences such as it follows that .
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: