The neighborhood of a function. Function sequence limit. MA. Function limit. Epsilon-Delta Definition Equivalence of Cauchy Function Limit Definitions
The general definition of a neighborhood of a point on the real line is considered. Definitions of epsilon neighborhoods, left-handed, right-handed, and punctured neighborhoods of endpoints and at infinity. Neighborhood property. A theorem on the equivalence of using an epsilon neighborhood and an arbitrary neighborhood in the definition of the Cauchy limit of a function is proved.
ContentDetermination of the neighborhood of a point
A neighborhood of a real point x 0
Any open interval containing this point is called:
.
Here ε 1
and ε 2
are arbitrary positive numbers.
Epsilon - neighborhood of point x 0
is called the set of points, the distance from which to the point x 0
less than ε:
.
The punctured neighborhood of the point x 0
is called the neighborhood of this point, from which the point x itself has been excluded 0
:
.
Neighborhood endpoints
At the very beginning, the definition of a neighborhood of a point was given. It is designated as . But you can explicitly specify that a neighborhood depends on two numbers using the appropriate arguments:
(1)
.
That is, a neighborhood is a set of points belonging to an open interval.
Equating ε 1
to ε 2
, we get epsilon - neighborhood:
(2)
.
Epsilon - a neighborhood - is a set of points belonging to an open interval with equidistant ends.
Of course, the letter epsilon can be replaced by any other and we can consider δ - neighborhood, σ - neighborhood, and so on.
In the theory of limits, one can use the definition of a neighborhood based both on the set (1) and on the set (2). Using any of these neighborhoods gives equivalent results (see ). But the definition (2) is simpler, therefore, it is epsilon that is often used - the neighborhood of a point determined from (2).
The concepts of left-handed, right-handed, and punctured neighborhoods of endpoints are also widely used. We present their definitions.
Left-hand neighborhood of a real point x 0
is the half-open interval located on the real axis to the left of x 0
, including the dot itself:
;
.
Right-hand neighborhood of a real point x 0
is the half-open interval located to the right of x 0
, including the dot itself:
;
.
Punctured Endpoint Neighborhoods
Punctured neighborhoods of the point x 0 are the same neighborhoods, from which the point itself is excluded. They are identified with a circle above the letter. We present their definitions.
Punctured neighborhood of point x 0
:
.
Punctured epsilon - neighborhood of point x 0
:
;
.
Punctured left-hand neighborhood:
;
.
Punctured right-hand neighborhood:
;
.
Neighborhoods of points at infinity
Along with the end points, the notion of a neighborhood of points at infinity is also introduced. They are all punctured because there is no real number at infinity (at infinity is defined as the limit of an infinitely large sequence).
.
;
;
.
It was possible to determine the neighborhoods of infinitely distant points and so:
.
But instead of M, we use , so that a neighborhood with a smaller ε is a subset of a neighborhood with a larger ε , just like for neighborhoods of endpoints.
neighborhood property
Next, we use the obvious property of the neighborhood of a point (finite or at infinity). It lies in the fact that neighborhoods of points with smaller values of ε are subsets of neighborhoods with larger values of ε . We present more rigorous formulations.
Let there be a finite or infinitely distant point. Let it go .
Then
;
;
;
;
;
;
;
.
The converse assertions are also true.
Equivalence of definitions of the limit of a function according to Cauchy
Now we will show that in the definition of the limit of a function according to Cauchy, one can use both an arbitrary neighborhood and a neighborhood with equidistant ends .
Theorem
The Cauchy definitions of the limit of a function that use arbitrary neighborhoods and neighborhoods with equidistant ends are equivalent.
Proof
Let's formulate first definition of the limit of a function.
A number a is the limit of a function at a point (finite or at infinity) if for any positive numbers there exist numbers depending on and , such that for all , belongs to the corresponding neighborhood of the point a :
.
Let's formulate second definition of the limit of a function.
The number a is the limit of the function at the point , if for any positive number there exists a number depending on , such that for all :
.
Proof 1 ⇒ 2
Let us prove that if the number a is the limit of the function by the 1st definition, then it is also the limit by the 2nd definition.
Let the first definition hold. This means that there are such functions and , so for any positive numbers the following holds:
at , where .
Since the numbers and are arbitrary, we equate them:
.
Then there are functions and , so that for any the following holds:
at , where .
Notice, that .
Let be the smallest positive number and . Then, as noted above,
.
If , then .
That is, we found such a function , so that for any the following is true:
at , where .
This means that the number a is the limit of the function and by the second definition.
Proof 2 ⇒ 1
Let us prove that if the number a is the limit of the function by the 2nd definition, then it is also the limit by the 1st definition.
Let the second definition hold. Take two positive numbers and . And let be the smallest of them. Then, according to the second definition, there is such a function , so that for any positive number and for all , it follows that
.
But according to . Therefore, from what follows,
.
Then for any positive numbers and , we have found two numbers , so for all :
.
This means that the number a is also the limit by the first definition.
The theorem has been proven.
References:
L.D. Kudryavtsev. Course of mathematical analysis. Volume 1. Moscow, 2003.
texvc
- neighborhood set in functional analysis and related disciplines is such a set, each point of which is removed from the given set by no more than Unable to parse expression (executable file texvc
not found; See math/README for setup help.): \varepsilon
.
Definitions
- Let be Unable to parse expression (executable file
texvc
not found; See math/README for setup help.): (X,\varrho) is a metric space, Unable to parse expression (executable filetexvc
not found; See math/README for setup help.): x_0 \in X, and Unable to parse expression (executable filetexvc
not found; See math/README for setup help.): \varepsilon > 0. Unable to parse expression (executable filetexvc
not found; See math/README for setup help.): \varepsilon-neighborhood Unable to parse expression (executable filetexvc
is called a set
texvc
not found; See math/README for setup help.): U_(\varepsilon)(x_0) = \( x\in X \mid \varrho(x,x_0)< \varepsilon \}.
- Let a subset be given Unable to parse expression (executable file
texvc
not found; See math/README for setup help.): A \subset X. Then Unable to parse expression (executable filetexvc
not found; See math/README for setup help.): \varepsilon-neighborhood of this set is called the set
texvc
not found; See math/README for setup help.): U_(\varepsilon)(A) = \bigcup\limits_(x \in A) U_(\varepsilon)(x).
Remarks
- Unable to parse expression (executable file
texvc
not found; See math/README for setup help.): \varepsilon-neighborhood of a point Unable to parse expression (executable filetexvc
not found; See math/README for setup help.): x_0 thus called an open ball centered at Unable to parse expression (executable filetexvc
not found; See math/README for setup help.): x_0 and radius Unable to parse expression (executable filetexvc
not found; See math/README for setup help.): \varepsilon. - It follows directly from the definition that
texvc
not found; See math/README for setup help.): U_(\varepsilon)(A) = \( x\in X \mid \exists y\in A\; \varrho(x,y)< \varepsilon\}.
- Unable to parse expression (executable file
texvc
not found; See math/README for setup help.): \varepsilon-neighborhood is a neighborhood and, in particular, an open set.
Examples
Write a review on the article "Epsilon neighborhood"
An excerpt characterizing the Epsilon neighborhood
- Well, what - listen? The little girl pushed me impatiently.We came close... And I felt a wonderfully soft touch of a sparkling wave... It was something incredibly gentle, surprisingly affectionate and soothing, and at the same time, penetrating into the very "depth" of my surprised and slightly wary soul... Quiet “music” ran along my foot, vibrating in millions of different shades, and, rising up, began to envelop me with something fabulously beautiful, something beyond words ... I felt that I was flying, although there was no flight was not real. It was wonderful!.. Each cell dissolved and melted in the oncoming new wave, and the sparkling gold washed right through me, taking away everything bad and sad and leaving only pure, primordial light in my soul...
I did not even feel how I entered and plunged into this sparkling miracle almost with my head. It was just incredibly good and I never wanted to leave there ...
- All right, that's enough already! We have a job ahead of us! Stella's assertive voice broke into the radiant beauty. - Did you like it?
- Oh, how! I breathed. - I didn't want to go out!
- Exactly! So some “bath” until the next incarnation ... And then they don’t come back here anymore ...
What icons besides inequality signs and modulus do you know?
From the course of algebra, we know the following notation:
- the universal quantifier means - "for any", "for all", "for each", that is, the entry should be read "for any positive epsilon";
– existential quantifier, – there is a value belonging to the set of natural numbers.
- a long vertical stick is read like this: “such that”, “such that”, “such that” or “such that”, in our case, obviously, we are talking about a number - therefore “such that”;
- for all "en" greater than ;
- the sign of the modulus means the distance, i.e. this entry tells us that the distance between values is less than epsilon.
Determining the Limit of a Sequence
Indeed, let's think a little - how to formulate a rigorous definition of a sequence? ... The first thing that comes to mind in the light of a practical lesson is: “the limit of a sequence is the number to which the members of the sequence approach infinitely close.”
Okay, let's write the sequence:
It is easy to see that the subsequence approaches the number -1 infinitely close, and the even-numbered terms approach "one".
Maybe two limits? But then why can't some sequence have ten or twenty of them? That way you can go far. In this regard, it is logical to assume that if a sequence has a limit, then it is unique.
Note: the sequence has no limit, but two subsequences can be distinguished from it (see above), each of which has its own limit.
Thus, the above definition turns out to be untenable. Yes, it works for cases like (which I did not quite correctly use in simplified explanations of practical examples), but now we need to find a strict definition.
Attempt two: “the limit of a sequence is the number that ALL members of the sequence approach, except perhaps for a finite number of them.” This is closer to the truth, but still not entirely accurate. So, for example, in a sequence, half of the members do not approach zero at all - they are simply equal to it =) By the way, the "flashing light" generally takes two fixed values.
The formulation is not difficult to clarify, but then another question arises: how to write the definition in mathematical terms? The scientific world struggled with this problem for a long time, until the situation was resolved by the famous maestro, who, in essence, formalized classical mathematical analysis in all its rigor. Cauchy proposed to operate with neighborhoods, which significantly advanced the theory.
Consider some point and its arbitrary neighborhood:
The value of "epsilon" is always positive, and, moreover, we are free to choose it ourselves. Suppose that in a given neighborhood there is a set of members (not necessarily all) of some sequence . How to write down the fact that, for example, the tenth term fell into the neighborhood? Let it be on the right side of it. Then the distance between the points and should be less than "epsilon": . However, if the “x tenth” is located to the left of the “a” point, then the difference will be negative, and therefore the module sign must be added to it: .
Definition: a number is called the limit of a sequence if for any of its neighborhoods (previously chosen) there is a natural number - SUCH that ALL members of the sequence with higher numbers will be inside the neighborhood:
Or shorter: if
In other words, no matter how small the value of "epsilon" we take, sooner or later the "infinite tail" of the sequence will FULLY be in this neighborhood.
So, for example, the "infinite tail" of the sequence will FULLY go into any arbitrarily small -neighborhood of the point. Thus, this value is the limit of the sequence by definition. I remind you that a sequence whose limit is zero is called infinitely small.
It should be noted that for the sequence it is no longer possible to say “an infinite tail will enter” - the terms with odd numbers are in fact equal to zero and “do not go anywhere” =) That is why the verb “end up” is used in the definition. And, of course, the members of such a sequence as also "do not go anywhere." By the way, check if the number will be its limit.
Let us now show that the sequence has no limit. Consider, for example, a neighborhood of the point . It is quite clear that there is no such number, after which ALL members will be in this neighborhood - odd members will always "jump" to "minus one". For a similar reason, there is no limit at the point .
Prove that the limit of the sequence is zero. Specify the number after which all members of the sequence are guaranteed to be inside any arbitrarily small -neighborhood of the point .
Note: for many sequences, the desired natural number depends on the value - hence the notation.
Solution: consider an arbitrary -neighborhood of the point and check if there is a number - such that ALL terms with higher numbers will be inside this neighborhood:
To show the existence of the required number , we express in terms of .
Since for any value "en", then the modulus sign can be removed:
We use the "school" actions with inequalities, which I repeated in the lessons Linear inequalities and Domain of definition of a function. In this case, an important circumstance is that "epsilon" and "en" are positive:
Since we are talking about natural numbers on the left, and the right side is generally fractional, it needs to be rounded:
Note: sometimes a unit is added to the right for reinsurance, but in fact this is an overkill. Relatively speaking, if we also weaken the result by rounding down, then the nearest suitable number (“three”) will still satisfy the original inequality.
And now we look at the inequality and recall that initially we considered an arbitrary -neighborhood, i.e. "epsilon" can be equal to any positive number.
Conclusion : for any arbitrarily small -neighborhood of the point, a value was found such that the inequality holds for all larger numbers. Thus, a number is the limit of a sequence by definition. Q.E.D.
By the way, from the result obtained, a natural pattern is clearly visible: the smaller the -neighborhood, the greater the number after which ALL members of the sequence will be in this neighborhood. But no matter how small the "epsilon" is, there will always be an "infinite tail" inside, and outside - even a large, but finite number of members.