The important functions of several variables. Basic understanding.

If each pair of independent numbers (x, y) from any multiplier is given the same rule as one value of the variable z, then it is called the function of two exchangeable. z = f (x, y,)

Function area z- The set of pairs (x, y) for which the function z is valid.

The impersonal value (range of value) of a function is all the values ​​that a function acquires in its domain of meaning.

Graph of the function of two changeable - an impersonal point P, the coordinates of which satisfy the level z = f (x, y)

Around the point M0 (x0; y0) to radius r- The totality of all points (x, y) that satisfy the mind< r

The area of ​​significance is the area of ​​the value of the function of several variables. Graph of the function of several variables.

Between and continuity of function of many variables.

Between the functions of several variables

To understand the inter-functions of several variables, separated by two Xі at. For the assigned function f(x, y) May be between exactly ( X 0 , at 0), equal to number A, which is designated as follows:

(1)

(Write more f(x, y)→A at (x, y)→ (X 0 , at 0)), since it is designated in the current vicinity of the point ( X 0 , at 0), behind the culprit, perhaps, the very points and the main boundaries

(2)

If only there had been no such thing as fire ( X 0 , at 0) sequence of dots ( x k ,y k).

So, just because there is no function of one variable, you can enter another equivalent value between the functions of two variable: function f may be exactly ( X 0 , at 0) boundary, level A, since it is designated in the vicinity of the point ( X 0 , at 0) behind the culprit, perhaps, the very same points, and for any ε > 0 there will also be δ > 0, so

| f(x, y) – A| < ε (3)

for everyone (x, y) what satisfies unevenness

0 < < δ. (4)

This value, in its own way, is equivalent to the next one: for any ε > 0 there is a δ-circle of the point ( X 0 , at 0) so that for everyone ( x, y) from this price area, from the front view ( X 0 , at 0) uncertainty is added (3).

Fragments of the coordinates of a significant point ( x, y) around the point ( X 0 , at 0) you can write it down x = x 0 + Δ X, y = y 0 + Δ at, then jealousy (1) is equivalent to offensive jealousy:

Let's look at the function given in the vicinity of the point ( X 0 , at 0), perhaps the most important points.

Let ω = (ω X, ω at) – additional vector of up to one (|ω|2 = ω X 2 + ω at 2 = 1) ta t> 0 – scalar. Specks of mind

(X 0 + tω X, y 0 + tω at) (0 < t)

confirm that you need to leave ( X 0 , at 0) directly from the vector ω. For the skin you can see the function

f(X 0 + tω X, y 0 + tω at) (0 < t< δ)

as a scalar variable t, de δ - Add a small number.

Between these functions (one variable t)

f(X 0 + tω X, y 0 + tω at),

As it turns out, it is natural to call it a boundary f at the point ( X 0 , at 0) directly ω.

butt 1. Functions

marked on the plane ( x, y) behind the point X 0 = 0, at 0 = 0. Maєmo (vrahuvati, scho і ):

(For ε > 0 it is important that δ = ε/2 and then | f(x, y)| < ε, если < δ).

This shows that between φ at the point (0, 0) along different straight lines there is a difference (a single vector of exchange y = kx, X> 0, looks like

).

Number A called the boundary of the function f(M) at M → M 0 if for any number ε > 0 there will always be a number δ > 0 such that for any points M, important views M 0 and satisfy the mind | MM 0 | < δ, будет иметь место неравенство |f(M) – A | < ε.

Meaning between There are two different functions

Theorems about boundaries. What are the functions? f 1 (M)і f 2 (M) at M → M 0 bend the skin to the end boundary, then:

V)

Continuity of function of several variables

For the assigned function f(x, y) uninterrupted exactly ( X 0 , at 0), since it is indicated in the current vicinity, including at the very point ( X 0 , at 0) and where the boundary f(x, y) This point has a similar meaning to it:

(1)

Mindfulness without interruption f at the point ( X 0 , at 0) can be written in equivalent form:

(1")

tobto. function f uninterrupted exactly ( X 0 , at 0), since the function is non-interruptible f(x 0 + Δ X, at 0 + Δ y) types of changes Δ X, Δ at at Δ X = Δ y = 0.

You can enter an increase Δ і functions і = f(x, y) at the point (x, y), which indicates an increase in Δ X, Δ at arguments

Δ і = f(x + Δ X, at + Δ y) – f(x, y)

And my main value is continuity f V (x, y): function f uninterrupted to the point (x, y), yakscho

(1"")

Theorem. Amount, difference, firmness and privacy of the uninterrupted at the point ( X 0 ,at 0) function f And φ is a non-stop function at this point, especially when it is private φ ( X 0 , at 0) ≠ 0.

Postinu h can be seen as a function f(x, y) = h kind of changeable x,y. She's constantly watching over these people, that's why

|f(x, y) – f (X 0 , at 0) | = |s – s| = 0 0.

Footing for foldability and functions f(x, y) = Xі f(x, y) = at. They can also be considered as functions of (x, y) And under this the stench is unceasing. For example, function f(x, y) = X bring the appearance of the skin point (x, y) the number that is older X. Continuity of this function at most points (x, y) can be done like this:

| f(x + Δ X, at + Δ y) – f(x, y) | = |f(x + Δ x) - x| = | Δ X | ≤ 0.

How to work on functions x, y and constant action of folding, visible and multiplying in the final number, then we can remove the functions called rich members in the form x, y. On the platform of the formulated higher authorities, there are a large number of important x, y– continuous functions from these critical points (x, y) R 2 .

Statue P/Q two rich members per (x, y) It is a rational function (x, y), obviously, uninterrupted throughout R 2, behind the vinyatok dot (x, y), de Q(x, y) = 0.

P(x, y) = X 3 – at 2 + X 2 at – 4

you can butt the butt of a rich penis (x, y) third stage, and the function

P(x, y) = X 4 – 2X 2 at 2 +at 4

є butt of a rich penis (x, y) fourth stage.

Let us introduce the theorem, which confirms the non-interruption of the function of non-interruptible functions.

Theorem. Let the function go f(x, y, z) uninterrupted to the point (x 0 , y 0 , z 0 ) space R 3 (points (x, y, z)), and the functions

x = φ (u, v), y= ψ (u, v), z= χ (u, v)

uninterruptedly to the point (u 0 ,v 0 ) space R 2 (points (u, v)). Let it go, besides,

x 0 = φ (u 0 ,v 0 ), y 0 = ψ (u 0 ,v 0 ), z 0 = χ (u 0 ,v 0 ) .

This function F(u, v) = f[ φ (u, v),ψ (u, v),χ (u, v)] uninterrupted (for

(u, v)) at the point (u 0 ,v 0 ) .

Finished. The boundary sign can be added under the sign of the characteristics of a non-interruptible function, then

Theorem. Function f(x, y), uninterrupted exactly ( X 0 , at 0) and not equal to zero at this point, preserves the sign of the number f(X 0 , at 0) in the current vicinity of the point ( X 0 , at 0).

For the assigned function f(x) = f(x 1 , ..., x p) uninterrupted to the point X 0 =(X 0 1 , ..., X 0 d) as it is indicated in the current surroundings, including at the very point X 0 and where between and exactly X 0 old meaning in it:

(2)

Mindfulness without interruption f at the point X 0 can be written in equivalent form:

(2")

tobto. function f(x) uninterrupted to the point X 0, since the function is non-interruptible f(x 0 +h) view h at the point h = 0.

You can enter an increase f at the point X 0, which indicates an increase h = (h 1 , ..., h p),

Δ h f (x 0 ) = f (x 0 + h) – f(x 0 )

And my meaning is continuity f V X 0: function f uninterrupted in X 0 , yakscho

Theorem. Sum, rіznitsa, tvir and privacy of the uninterrupted at the point X 0 functions f(x) ta φ (x)є uninterrupted function at this point, as, of course, at the same time φ (X 0 ) ≠ 0.

Respect. Gain Δ h f (x 0 ) also called enhanced functions f at the point X 0 .

In space Rn dot X = (x 1 , ..., x p) set a dead point G.

For appointments X 0 = (X 0 1 , ..., X 0 d)є internal point of the multiplier G, as it is clear that the core is open with a center in the new place, which will remain in place until G.

Bezlich G Rn is called open because all its internal points.

Say what the functions are

X 1 = φ 1 (t), ..., x n =φ p(t) (a ≤ t ≤ b)

without interruption for cutting [ a, b], denote a continuous curve in Rn, which connects the points X 1 = (X 1 1 , ..., X 1 d)і X 2 = (X 2 1 , ..., X 2 d), de X 1 1 = φ 1 (A), ..., X 1 n =φ p(a), X 2 1 = φ 1 (b), ..., X 2 n =φ p(b). Litera t called the curve parameter.

6.1. Between and continuity of function of many variables.

R n – metric space:

For M 0 (x, x,…, x) that M(X 1 , X 2 , …, X n) ( M 0 , M) = .

n= 2: for M 0 (x 0 , y 0), M (x, y) ( M 0 , M) =
.

Outskirts of the point M 0 U  (M 0) = – internal points of the stake radius centered at M 0 .

6.1.1. Between the functions of several variables. Repeated boundaries.

f: R n R specified in the current vicinity of the point M 0 , cream, perhaps, most points M 0 .

Viznachennya. Number A called boundary functions

f(x 1 , x 2 , …, x n) at the point M 0 , yakscho  >0  >0  M (0 <  (M 0 , M ) <   | f (M ) – A |<  ).

F ormi note:

n = 2:

Tse boundary line.

There's a point around me:

>0  >0  M (x , y ) (M  U  (M 0 )\ M 0  f (x , y )  U  (A )).

(M we can get closer to M 0 whatever way).

Repeated boundaries:
і
.

(M approaching M 0 horizontally and vertically).

Theorem about the connections between the sub-links and repeated interconnections.

What is the boundary line?
і boundaries
,
,

then repeat the boundaries
,
and be like the subordinate.

Respect 1. The turning point is incorrect.

butt. f (x, y) =

,

.

However, the boundary

=

It doesn’t matter, because in any neighborhood of the point (0, 0) the function accepts “far” values ​​​​from zero, for example, as x = y, That f (x, y) = 0,5.

Note 2. Tell me how  AR: f (x, y) A

under Russia M before M 0 there may be no need to sleep behind any straight, curved boundary.

butt.f (x, y) =
,M 0 (0, 0). M (x, y)  M 0 (0, 0)

Concept: the boundary (sub-surface) does not exist.

Butt marking of boundaries.

f (x, y) =
, M 0 (0, 0).

Let us show that the number 0 is between the functions of the point M 0 .

=
,

 – stand between the points Mі M 0 .(accelerated by nervousness
,

how it escapes from inconveniences
)

Assign  > 0 and let  = 2. <  

6.1.2. Continuity of function of several variables.

Viznachennya. f (x, y) is uninterrupted exactly M 0 (x 0 , y 0), since it was designated for action U  (M 0) that
,T. e.>0 >0  M (0 < (M 0 , M) <   | f (M) – f (M 0)|< ).

Respect. The function can be changed continuously along the same lines that pass through the point M 0, and in other directions and ways of other forms of mother development. That's right, it's razrivna at the point M 0 .

6.1.3. The power between the functions of many different ones. The power of functions, uninterrupted at the point.

May be the place Unity between boundaries;

function that draws the end boundary exactly M 0 , surrounded by this point; concur order and algebraic powers boundaries,

border crossing preserves signs of jealousy and mild anxieties.

Since the function is uninterrupted at the point M 0 that f (M 0 )  0 , That sign meaningf (M ) is saved in action U  (M 0).

Suma, tvr, private(sign  0) without interruption functions as well uninterrupted functions, continuous folding function, Folded without interruptions.

6.1.4. The power of functions that are uninterrupted on a connected, closed, bounded multiplicity.n= 1, 2 and 3.

Value 1. Bezlich  is called viscous How to simultaneously place two points of your own and create a continuous curve that connects them.

Value 2. Bezlic  in R n called let's circumcise How to fit into this “cool”
.

n = 1 

n = 2 

n = 3  .

Apply itviscous closed boundaries of multiplicities.

R 1 = R: video [ a, b];

R 2: video AB be some kind of continuous curve with ends at points Aі U;

the curve is closed without interruption;

colo
;

Value 3. f: R n  R continuous on a connected closed multiplicity   R n, yakscho  M 0 

.

Theorem.Bezlichmeaning uninterrupted functions

f: R n  R on a closed boundary of a connected multiply and a section [ m , M ] , here m - smaller, A M - the greatestїї values ​​​​at the multiplying points.

In such a manner on any closed boundary of a connected multiplicity inR n The continuous function is circumscribed, takes on its lowest, highest, and all intermediate values.

Between the functions of two changeable ones.
Understand and apply it

We kindly ask for the third lesson on the topic FNP, where all your worries have begun to come true =) As many have suspected, the understanding between them is expanding and to the function of quite a number of arguments, which we need to understand today. Prote is an optimistic novelty. The point is that between the world of abstraction and everyday tasks, the edges rarely meet in practice. Our respect will be divided between the functions of two important characters, as is often written: .

There are a lot of ideas, principles and methods similar to the theory and practice of “primary” ones between, also, Narazi it's your fault note the boundaries And most importantly, UNDERSTAND what it is between functions of the same variable. And since fate brought you to this side, then, having seen everything, you already understand. And if they don’t – there’s nothing terrible, all the clearings can actually be filled for treatment and healing.

The ideas of this activity are burning in our trivial world, and it would be simply a great oversight not to take a living part in them. From now on we will know better Cartesian coordinate system in space. Let's get up and walk around the room for a while... ... the reason you walk is flat. Let’s put everything here... well, for example, if anyone has a corner, so that they don’t care about the road. Miraculous. Now, kindly, marvel at the mountain and see that there is a carpet hanging there. Tse surface, is specified by the function. Our movements along the underside, as is easy to understand, involve a change of unimportant changes, and we can move around, including under the carpet, then. V areas of significant function of two important. Once again it just begins. Just above the tip of your nose, a small targan is hovering across the carpet, wherever you go. Let's call him Freddy. This movement changes the corresponding function values (besides these fallouts, if the surface or fragments are parallel to the plane, the height does not change). I’m reading from Freddy, don’t act like that, it’s required for science.

We take an awl in our hands and pierce the carpet at a sufficient point, the height of which is significant through, after which we place the tool under the opening at the base - this will be the point. Now let's start incredibly close approach this point , and we have the right to approach BEHIND ANY trajectory (The skin point which, apparently, enters the designated area). How will Freddy have in all his episodes? incredibly close raise the height to the puncture and the same height, then the function runs between the points at :

As for the indications of their minds, a point is pierced and slashed on the edge of the carpet, then between all one thing is indispensable - it is important that How many small neighborhoods are there each year? The window sewed any points in the area of ​​the assigned function. In addition, as I have a problem with between the functions of one changeable, doesn't matter, which is designated by the function at the point of ni. So our puncture can be sealed with a bug (rahuvati, sho the function of the two exchangeable ones is uninterrupted) Regardless of the situation, it is clear that the very essence of boundaries is respected infinitely close proximity, and not an “exact approach” to the speck.

However, a gloomy life is overshadowed by the fact that in the presence of his young brother, he often does not sleep. This is connected with this, that to this other point on the plane there are already a lot of nobles, and the skin of them is responsible for bringing Freddie to the puncture. (optional “stuck with a bug”) and strictly at height. And on the chimeric surfaces with no less chimerical ripples I want to row the roads, which will lead to the destruction of this fierce mind at certain points.

We organize the simplest butt- We take the knife in our hands and cut the carpet in such a way that the pierced point lies on the cut line. Respect the boundaries It is still true, united, that we have lost the right to step into the spots under the line of the cut, as a result of which this plot “fell” from areas of significant function. Now carefully lift the left part of the carpet up the axis, and the right part, for example, crush it down or place it firmly in place. What has changed? And the principle has changed like this: as soon as we are approaching the point of evil, then Freddy will appear at a higher altitude, lower, as if we were approaching this point right-handed. Well, there are no boundaries.

I, of course, miracle boundaries go without them. Let’s take a look at the common butt of all senses:

Butt 11

Vikorist can use the painfully familiar trigonometric formula, which is organized using a standard piecemeal technique first miracle boundaries :

Let's move on to polar coordinates:
Something like that

It would seem that the decision is going to a natural outcome and there is no sense of inconsistency, but there is, however, a great risk of allowing a serious shortcoming, about the nature of which I have already drawn up a little in Appendix 3 and wrote a report after Appendix 6 . First ending, then comment:

Let’s figure out why it’s bad to write down simply “inconsistency” or “plus inconsistency.” Marvel at the banner: the fragments, then the polar radius of the infinitely small Positively meaningless: . In addition, . In this way, the sign of the banner and all boundaries lie only within the cosine:
Yakshcho Polar Kut (2nd and 3rd coordinate quarters: );
Yakshcho Polar Kut (1st and 4th coordinate quarters: ).

Geometrically, this means that if you approach the beginning of the coordinates of the surface, then the surface is specified by the function , extends endlessly down:

The above-mentioned concepts of the functions of two or three variables can be identified in different variables.

Viznachennya. function significant
called function, designated area
which one is due
, And the area of ​​​​value is the action axis.

This is the function of the skin set
h
leaves one .

Below we look at the importance of the functions
changes, but all the statements formulated for such functions are no longer valid for functions with a large number of changes.

Viznachennya. Number called the boundary of the function

at the point
, for skin
there is such a number
what for everyone
from the outskirts
, besides these points, inequality is drawn

.

Between the functions
at the point
more ancient , then this is indicated in the view

.

Almost all the powers between the functions we discussed earlier for the function of one change will be deprived of fairness and between the functions of many changes, we will not deal with the practical exchanges of such between us.

Viznachennya. Function
is called uninterrupted exactly
There are three minds that come together:

1) is asleep

2) the main significance of the function of a point

3) these two numbers are equal to each other, then. .

It is possible to practically verify the continuity of a function using an additional theorem.

Theorem. Whether the function is elementary
is continuous at all internal (that is, not border) points of its assignment area.

butt. We know all the points that have a function

uninterrupted

As the thing was assigned, this function is assigned to a closed circuit

.

The internal points of this stake are points of continuity of function, then. function
uninterrupted in open air
.

The meaning of the concept of continuity at the boundary points of the area of ​​​​significance
The functions are possible, but we will not study the nutrition in the course.

1.3 Private additions and private additions

By substituting the function of one variable, the functions of several variable ones may result in different types of increments. This is due to the fact that displacements near the plane
from the point
can be done from different directions.

Viznachennya. Private extras for functions
at the point
to the latest increases
called difference

This is essentially a greater function of one change
removed from the function
at steady value
.

Similar to private benefits at the point
functions
to the latest increases
called difference

More is calculated when the value is fixed
.

butt. Let's go

,
,
. We know the private increased price of functions for and by

In which case, with equal values, the increment of arguments
і
, the private functions were different. This is due to the fact that the area of ​​the rectangular plant is the sides
і
on the larger side on
increases by amount
, and with larger sides on
increases by
(div. Fig. 4).

From the fact that the function of two variables has two types of increment, it follows that two types of increments can be assigned to it.

Viznachennya. Private campaign for functions
at the point
is called between the relations of private increase in These functions are at the indicated point until increased
argument tobto.

. (1)

Such expeditions are indicated by symbols ,,,. In the remaining episodes of the round of letters “ ” – “” means the word “private”.

Similarly, the privacy of the at the point
appears beyond the border

. (2)

Other private prices: ,,.

Private similar functions are subject to the following rules of differentiation of the function of one variable, and all changes, except those for which differentiation the function is subject to yinimi. So when you know zminna gets down to work, and at the hour of overwork - Postiina .

butt. We know the private secret functions
.

,
.

butt. We know the private functions of the three important ones

.

;
;
.

Private travel functions
characterize the speed of change of this function at a time, if one of the changes is fixed.

An example of economics.

The basic concepts of the theory of coexistence are the cost function
. This function expresses the amount of color of the set
, where x is the quantity of product X, y is the quantity of product U. Todi private company
will obviously be called the boundary values ​​x and y. Limit rate of substitution
one product and another of the same value of their border values:

. (8)

Task 1. Find the boundary rate of substitution of the year for the corisivity function at point A (3.12).

Decision: after formula (8) we remove

The economical substitution of the boundary norm of substitution lies in the circumscribed formula
, de -price of product X, - price of product U.

Viznachennya. What is the function
If they are private, then they are called private differentials.

і

here
і
.

Private differentials are differentials of the functions of one variable, separated from the functions of two variables.
when fixed or else .

Applications from economics. Let's take a look at the Cobb-Douglas function.

Magnitude - average productivity of the plant, including a large number of products (in the Vartisan expression), generated by one plant worker.

Magnitude
- Average capital output is the number of products that fall on one mile.

Magnitude
- Average capital-labor ratio is the number of funds that falls on one unit of labor resources.

That's why it's private
is called the marginal productivity of the product, the fragments of the traditional additional product yield, generated by another additional robotic worker.

Likewise,
- limited fund return.

In the economy, often put food: how many hundreds of hundreds of people will be required to produce products if the number of workers increases by 1% or if the capital increases by 1%? These types of nutrition give an understanding of the elasticity of the function behind the argument, which is obviously similar. We know the elasticity of product output from practice
. Substituting up to the number I will calculate more privately , cancelable
. Ozhe, parameter There is a clear economic sense - the elasticity of the issue is not in question.

A similar sensor parameter - the elasticity of the release of funds.

Let's take a look at the area and system Oxy Cartesian rectilinear coordinates on it (you can see other coordinate systems).

From analytical geometry we know that each ordering of a pair of numbers (x, y) you can align a single point M flatness and shape, skin points M The area is indicated by a single pair of numbers.

That's why, speaking about the point, we often respect this pair of numbers (x, y) And by chance.

Meaning 1.2 Unpersonal pairs of numbers (x, y) , which satisfies unevenness, is called a straight cutter (vkritim).

On the plane of the veins it will appear as a rectum (Fig. 1.2) with sides parallel to the coordinate axes, and with a center at the point M 0 (x 0 y 0 ) .

The straight cutter is commonly used as an offensive symbol:

Let us introduce the more important meaning of the far viklad: the outskirts of the point.

Value 1.3 Straight δ -outskirts ( delta outskirts ) points M 0 (x 0 y 0 ) called upright

with center at point M 0 and with equal sides 2δ .

Vicenza 1.4 Circular δ - around the point M 0 (x 0 y 0 ) called colo radius δ with center at point M 0 , then there is no point M(xy) the coordinates of which satisfy the unevenness:

It is possible to understand a number of other types of circuits, but through mathematical analysis of technical requirements, it is important to distinguish between straight and circular circuits.

Let us introduce the following concepts between the functions of the two.

Let the function go z = f(x, y) designated in deyaki galusi ζ і M 0 (x 0 y 0 ) - A point that lies in the middle or at the border of this area.

Value 1.5 Kintseve number A called boundary of the function f(x, y) at

yakscho for whatever positive number ε Is it possible to know this positive number? δ , what is nervousness

Configured for all points M(x,y) from the region ζ , important views M 0 (x 0 y 0 ) , the coordinates of which satisfy the inequalities:

The sense of significance lies in the significance of the function f (x, y) how the numbers A in the points vary a little every year to a small area around the point M 0 .

Here the basis of the design is laid out by straight-cut surroundings M 0 . It would be possible to see the circular surroundings of the point M 0 And then it would be necessary to overcome the conflicting inequalities

at all points M(x,y) regions ζ , important views M 0 and satisfy the mind:

Stand between the points M і M 0 .

The following boundaries are used:

Based on the significance of the interchange functions between two interchangeable ones, it is possible to transfer the main theorems about interchanges for the function of one interchangeable function and two interchangeable ones.

For example, theorems about boundaries involve parts of two functions.

§3 Non-interruption of the function of two changes

Let the function go z = f (x, y) indicated at the point M 0 (x 0 y 0 ) the same surroundings.

Value 1.6 The function is called continuous at a point M 0 (x 0 y 0 ) , yakscho

What is the function f(x,y) uninterrupted to the point M 0 (x 0 y 0 ) , That

Oskolki

What is the function f(x,y) uninterrupted to the point M 0 (x 0 y 0 ) , then this galusia suggests an infinitely small increase in arguments Δz functions z .

Fair and reversal assertions: if an infinitely small increase in arguments indicates an infinitely small increase in the function, then the function is uninterrupted

A function that is uninterrupted in the skin area is called uninterrupted in the area. For continuous functions of two changeables, like a function of one changeable, without interruption to a cut, the basic theorems of Weierstrass and Bolzano-Cauchy are valid.

Evidence: Karl Theodor Wilhelm Weierstrass (1815 - 1897) - German mathematician. Bernard Bolzano (1781 – 1848) – Czech mathematician and philosopher. Augustin Louis Cauchy (1789 - 1857) - French mathematician, president of the French Academy of Sciences (1844 - 1857).

butt 1.4. Check the continuity of the function

This function is defined for all variable values x і y , at the beginning of the coordinates, where the sign changes to zero.

rich member x 2 +y 2 is without interruption here, and therefore is without interruption a square root with a non-interruption function.

The flow will be uninterrupted everywhere, except the point where the sign is equal to zero. This function, as seen, is continuous throughout the entire coordinate plane Ohoo , including the coordinate cob.

butt 1.5. Check the continuity of the function z = tan (x, y) . The tangent of the values ​​of i is continuous for all terminal values ​​of the argument, including values ​​equal to the unpaired number of the value π/2 , then. including points, de

For skin fixed "k" equal (1.11) means hyperbole. Therefore, the function that is seen is a non-stop function x and y including points that lie on curves (1.11).