Real Analysis – Limits and Continuity IV
As an application of theorem 35 let us look into the functions and .
Now and is a strictly increasing function, and also is a strictly increasing function.
By theorem 35 it is and .
As for it is and .
If in the previous definition doesn’t equal zero:
 .
 .
 is bounded in some neighborhood of .
These notions work exactly as they worked for sequences and they give the same type of information about the behavior of the functions in question.
Theorem 36 Let ; , and . Then:
Proof: Left as an exercise. 
As an example of the previous definitions we can say, with full generality, that for any polynomial function we can keep track of the term with the leading degree if we are interested in how it behaves for larger and larger values.
But on the other hand if we are interested on how the polynomial function behaves near the origin we have to keep track of the term with the smaller degree. To see that this is indeed so let us introduce the following example:
Now . If we take it is and so it is .
Another example that has a lot of interest to us is:
We can see that it is so because of
— 6.6. Epsilondelta condition —
And it is time for us to introduce the concept of limit using the condition.
Once again we are walking into regions of greater and greater rigor at the expense of having to use more abstract concepts while we are doing it. Things are going to get a little harder for people that aren’t used to this types of reasoning but please bear with me and you’ll find it rewarding when you get used to it.
The point of the condition is to avoid using fuzzy concepts near, input signals, output signals, or the somewhat weak definition of limit we been using so far.
Theorem 37 (Heine’s Theorem)
Let , , and . if and only if
Proof: Omitted. 
In case you are wondering what that means the straightforward answer is that it means exactly what you’re idea of a function having a limit in a given point is (I’m assuming you have the right idea). It tell us that if a function indeed has limit in point then, if we restrict ourselves to points near , the images of those points are all near .
Once again I tell the reader to look at this as if it were a game played between two (slightly odd) people. One of them is choosing the and the the other is choosing the . But this game isn’t just about choosing. The first player gets to choose any he wants, but the second has to choose the right that makes the condition hold.
If he can prove that he has an for every that the other player chooses than he succeeds in the game and the function does have limit at point .
Theorem 38
Let , , and . If exists and is finite, than there exists a neighborhood of where is bounded. Proof: Let . By theorem 37 with there exists such as
Thus . So and is bounded in 
If exists, then since in this case it is and there exists some neighborhood of where is bounded.
After this one may be interested in knowing how we can translate to a condition.
In this case we are considering only in the set and so what we get is:
Theorem 39 Let , , and . If , then .
Proof: Let . By the condition it is:
Thus by taking it follows or In conclusion: which is equivalent to saying that . 
Definition 34 Let ; and . We say that is continuous in point if for all sequences of points in , such as it is .
A function is said to be continuous if it is continuous in all points in . 
A few examples to clarify definition 34

Let and a sequence such as . Then and . In conclusion which is equivalent to saying that is continuous in . Since can be any given point is continuous in .
 Let and a sequence such as . It is and by the same reasoning is also continuous.
 In general if it is . So for it is .
If it follows that and for it follows that .
Thus if we define and it follows that it always is .
 Analogously we can define and and it always is .
Theorem 40 (Heine’s theorem for continuity)
Let , and . is continuous in if and only if
Or written in terms of neighborhoods
Proof: Omitted. 
As can be seen the condition for continuity in point is very similar to the one for limit in point .
To finish this post I’ll just state a theorem that sheds some light on the connections of these two concepts:
Theorem 41 Let , and . Then it’s continuous in point if and only if .
Proof: Omitted. 
So as this theorem shows the connection between continuity and limit is indeed a deep one, but we can look at the concept of limit as being an auxiliary tool to determine if a function is continuous or not and we should not confuse them.
In the next post I intend to write a little bit more about continuity but in the mean time a very good text about it can be found here
April 19, 2011 at 5:04 am
Then a f is increasing on a b if f x 0 for each x a b b f is decreasing on a b if f x 0 for each x a b This theorem can be proved by using Mean Value Theorem. We shall prove the theorem after learning Mean Value Theorem.This theorem is applied in various problems to check whether a function is increasing or decreasing.