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Subido el 29 de julio de 2024 por Francisca F.

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We are going to see in this video how to study the continuity of a function 00:00:03
and above all we are going to see examples of functions defined in pieces. 00:00:07
When is a function continuous in a point? 00:00:12
A function is continuous in a point if these three conditions are met. 00:00:15
First, there has to be the limit of the function when x tends to that point. 00:00:20
Remember that in order for the limit of the function to exist in a point, 00:00:26
There must be the two lateral limits, the lateral limit on the right, 00:00:32
the lateral limit on the left, and coincide, and the result is a finite number. 00:00:41
We are going to call that limit L. On the other hand, the function must be defined at that point. 00:00:50
That is, that point A has to belong to the domain of the function. 00:00:57
In other words, there is F in A, there is the image of A. 00:01:04
And, on the other hand, the two values, the limit and the value of the function in the point, have to coincide. 00:01:15
That is, in this case, as we have set the limit L, 00:01:24
what the function in A is worth has to coincide with that value that we have obtained for the limit. 00:01:28
For a function to be continuous in an interval, it has to be continuous in all the points of the interval. 00:01:35
If it is continuous in all the points of the interval, it will be continuous in the interval. 00:01:44
And many times the expression is used, a function is continuous in all its domain. 00:01:49
What does that mean? 00:01:54
Well, that these conditions are fulfilled for all the points that make up their domain. 00:01:56
Look, this function f is a function defined in pieces. 00:02:02
Notice that for x less than or equal to 3, it is defined as a rational function. 00:02:06
This is a hyperbola in this case. 00:02:16
and for those greater than 3, it is also defined as another interval, as another rational function. 00:02:19
Here we have the graph of the function to understand what we are going to see, 00:02:26
but in reality they can ask me to study the continuity of the function and not give me the graph. 00:02:31
How would we do it? If they ask me to study the continuity of the function, 00:02:37
Well, the first thing we are going to do is to study each of the open intervals. 00:02:45
First from minus infinity to 3, see what happens, and from 3 to infinity. 00:02:51
Look at the function. 00:02:56
In the interval that goes from minus infinity to 3, the function is defined in this way, right? 00:02:59
And this function, we see that when it is a rational function, it is not defined for x equal to 2. 00:03:09
We also know that it is a hyperbola, that for x equal to 2 we have a vertical asymptote. 00:03:17
What we see is that x equal to 2 is in this interval. 00:03:22
Therefore, we know that here, in 2, the function will have that asymptote. 00:03:30
So, this function, 2 divided by x-2, is continuous in the interval from minus infinity to 2 union from 2 to 3. 00:03:38
In x equal to 2, the function is not defined, that is, there is no f in 2, the image of 2, 00:04:01
and the lateral limits on the right and on the left of the function are divergent. 00:04:09
In this case, the function, look, if I take a value close to 2 on the right, 2.000-2, 00:04:24
the one above is going to be positive, the one below too, this tends to be more infinite. 00:04:33
If you look at the graph, I can see perfectly that when I get closer to 2 on the right, 00:04:37
the function goes to plus infinity, in this asymptote that it has, in x equal to 2. 00:04:42
But taking limits I can also see it. 00:04:49
And the limit when x tends to 2 to the left, 00:04:52
in this case we would take a value close to 2 to the left that can be 1.999. 00:04:59
If we make that difference in the denominator, 00:05:05
will give us negative, positive between negative, negative, this limit tends to 00:05:09
minus infinity. And we see it in the graph here, that when I get closer to 2 on the left 00:05:14
I go to minus infinity. Well, let's analyze now the interval 00:05:19
that goes from 3 to infinity. The function, we said, 00:05:24
is also defined as a rational function and also 00:05:29
Notice how the degree of the numerator is 1 and the degree of the denominator is also a hyperbola. 00:05:34
Well, the function 3x divided by 2x minus 3 is not defined for what value of x? 00:05:41
Well, when x is 3 halves, which is when the denominator is nullified. 00:05:52
But what happens is that 3,5 is not in this interval from 3 to infinity, 00:05:57
but it is in the interval that goes from minus 3, that is, from minus infinity to 3. 00:06:04
This function is continuous in the interval from 3 to infinity. 00:06:09
In this interval it is continuous. 00:06:20
And we see it in the graph, in the graph, this piece of hyperbola is 00:06:25
for the x greater than 3, and we see that from 3 the function is contained. 00:06:31
It can be drawn from a single line. 00:06:36
And now let's see what happens in x equal to 3. 00:06:38
In x equal to 3, first we see what the function in 3 is worth, 00:06:43
we would have to look here, which is where it is defined, 00:06:49
and substituting for x equal to 3 I obtain the value 2. 00:06:52
The limit, when x tends to 3 on the right of the function, 00:06:58
of course, if I get closer to the right of the function, I have to take this expression for f 00:07:05
and substitute in it the x for 3, 00:07:11
then I would have 9 divided by 6 minus 3, 00:07:22
that is, this result would be 3. 00:07:26
And the limit, when x tends to 3 on the left, 00:07:29
In this case, if I get closer to 3 on the left, what expression do I have to take? 00:07:36
Well, I get closer to this other side, I have to take this one, for the f. 00:07:45
And substituting, the result is 2. 00:07:52
This, that we do here with the calculation, well, with the graph we saw it clearly, right? 00:07:57
If I get closer to 3 on the right, the images of these x are approaching the value of 3. 00:08:02
That's why the lateral limit on the right has given me 3. 00:08:10
On the left, we also saw it very well, 00:08:14
the images of these x are approaching the value of 2. 00:08:18
That's why this limit is worth 2. 00:08:23
And what is worth the function in 2 is defined. 00:08:26
You see that here we have a full point, so for 3 the function is worth 2. 00:08:28
But now we are assuming that we do not have the graph, 00:08:33
because we are doing everything with the mathematical expression of the function. 00:08:36
The most convenient thing is to put this diagram to see if I go to the right what function I have to place, 00:08:43
for the f what expression, if I go to the left what other expression I have to put. 00:08:49
In short, what do we see? That the function is not continuous in x equal to 3. 00:08:55
In x is equal to 3, the function is not continuous and the discontinuity it presents is not continuous. 00:09:02
The discontinuity it presents is finite jump because we see that the lateral limits are different. 00:09:16
So it is an inevitable discontinuity of finite jump. 00:09:23
We are going to see another example of continuity study and we are also going to add the derivability of the function. 00:09:27
First the continuity and then we are going to study the derivability of the function. 00:09:34
First we are going to see the intervals that we have. 00:09:41
The intervals that we have, notice that the function has like three sections. 00:09:45
The first one, from minus infinity to zero, is defined as x squared. 00:09:53
From zero to three, the function is defined as x. 00:09:59
And from three to infinity, it is defined as a constant function that is always valid. 00:10:02
Well, the first thing we see is what happens in the open intervals from minus infinity to zero. 00:10:10
In this interval, the function f is continuous, 00:10:17
the function y equal to x squared is continuous in all r. 00:10:24
So, in particular, it will be continuous in this open interval from minus infinity to zero. 00:10:30
From zero to three, the function f is also continuous in that open interval. 00:10:38
It is continuous because the function y equal to x is continuous in all r. 00:10:45
It has no discontinuity problem. 00:10:51
And from 3 to infinity, the function is defined as y equals 1, which is a constant function, 00:10:55
so it is continuous, since this function is continuous in all of R. 00:11:05
We still have to study what happens in x equals 0 and in x equals 3. 00:11:13
Let's start with x equals 0. 00:11:19
To see if the functions continue in x equal to 0, the first thing I have to see is what the function in 0 is worth. 00:11:24
The function in 0 is defined here, that the inequality appears with the equal, 00:11:30
less or equal, for x equal to 0 I will have to substitute here. 00:11:38
In this case, f in 0 is worth 0. 00:11:42
Now we are going to see the lateral limits on the right and on the left of 0. 00:11:48
If I approach 0 on the right, the expression I have to take is 00:11:56
y equals x, that is, here in the limit, instead of f , I will put x. 00:12:02
And when I substitute x for 0, this limit gives me 0. 00:12:10
And on the left, the limit when x tends to 0 on the left, 00:12:14
now I have to take this other expression, 00:12:22
When I substitute the x by 0, this limit gives me 0. 00:12:27
What do I see? 00:12:33
That these three things, the value of the function in the point, the lateral limits, 00:12:35
are finite and coincide. 00:12:40
So in x equal to 0, the function is continuous. 00:12:42
f is continuous. 00:12:49
Now we are going to see the point x equal to 1, x equal to 3. 00:12:56
In x is equal to 3, we do the same, we study the value of the function in 3. 00:13:03
In this case, notice that the function is not defined in x is equal to 3, 00:13:10
because here it puts a strict inequality, it puts x less than 3, 00:13:15
so here I can't substitute, but here neither. 00:13:19
So it doesn't exist, the function is not defined in 3. 00:13:23
Only for this reason the function would not be continuous, 00:13:29
But if we want to classify the discontinuity, we will have to see the lateral limits on the right and on the left. 00:13:32
We study the limit when x tends to 3 on the right of the function, and in this case we have to substitute, 00:13:41
we go to 3 on the right, we have to substitute here, in the function that is constant. 00:13:52
This limit gives me 1. 00:13:59
And the limit, when x tends to 3 on the left, 00:14:01
well, now I would have to take this other expression, the x, right? 00:14:05
And by substituting x by 3, I get 3. 00:14:15
Notice, the function does not exist in 3. 00:14:19
And we see that the lateral limits do exist. 00:14:23
They are convergent, one tends to 1, one is 1 and the other is 3. 00:14:27
Here what we see is that the lateral limits do not coincide. 00:14:31
So, what we have in x is equal to 3 is an inevitable discontinuity type. 00:14:45
And the jump of the discontinuity, notice that we go from having, from being 1 to being 3. 00:15:03
The jump of two units, right? Finite jump. 00:15:11
The limits have come out finite and the difference between them in absolute value gives me a finite jump. 00:15:15
So we have an inevitable discontinuity of finite jump. 00:15:22
Summing up, the function is continuous, let's put it here, f is continuous in all r 00:15:26
excepting the value x is equal to 3, where the function, in addition to not being defined, 00:15:40
We have seen that the lateral limits do not coincide, there is an inevitable discontinuity of the finite jump. 00:15:50
Well, seen first the continuity, we are going to see now the derivability of the function. 00:15:57
We are going to study where the function is derivable. 00:16:03
We have to derive the function and we are going to derive each of these expressions. 00:16:16
The derivative of x squared is 2x, this would be defined for the x less than 0. 00:16:25
The derivative of x is 1. 00:16:33
Notice that here what we are going to put is going to be the open interval, okay? 00:16:37
We are going to write the two inequalities without the equal. 00:16:43
Here was the equal, here we are not going to put it, now we explain why. 00:16:47
And the derivative of 1 is 0. 00:16:51
That is, I am deriving the function into pieces, section by section, 00:16:55
and if in any of the inequalities an equal appears, here we are going to remove it. 00:17:00
We are going to remove it because then we are going to see what happens in x is equal to 0 and in x is equal to 3 00:17:05
and the derivative will exist if the lateral derivatives in those points exist and coincide. 00:17:11
Therefore, we a priori here we have to write the function with the strict inequalities, without equal. 00:17:18
Well, another important thing is that the function where the function is not continuous 00:17:25
will not be derivable, obviously, so that's why we say that before you have to study 00:17:33
the continuity and we said before that in x equal to 3 the function f of x is not continuous 00:17:39
So, we already know that the function in x is equal to 3 is not derivable, there is no derivative in 3. 00:17:47
And it is logical, right? In x is equal to 3, if the function is not even defined, 00:18:06
but if it is not even continuous, I cannot establish a tangent line to the curve at that point, 00:18:12
then it will not be derivable. 00:18:20
Let's go step by step, as we have done before in the continuity. 00:18:25
In the interval that goes from minus infinity to zero, the function f is going to be derivable. 00:18:29
From zero to three, the function is defined as x and it is derivable in all the points. 00:18:45
Remember, this is the function f , in all these points I can draw the tangent line of the curve in those points. 00:18:56
And in fact here, since it is a line, the tangent line would coincide with the same line. 00:19:03
So f is derivable in that interval. 00:19:08
And from 3 to infinity, the function was defined with a constant, right? It's always worth 1. 00:19:16
I can always draw the tangent line in each of these points, so the function is derivable. 00:19:25
So it is derivable in the open intervals from minus infinity to zero, from zero to three, and from three to infinity. 00:19:34
What happens in zero and what happens in three? 00:19:46
Well, in three we have already said that since the function is not continuous, it is not derivable. 00:19:49
We have already committed this. 00:19:54
Let's see what happens in 0. 00:19:56
In x equal to 0, we study the derivative on the right of 0 and on the left. 00:20:03
If these two lateral derivatives exist and coincide, then the function will be derivable in 0. 00:20:13
In 0, if I approach from the right, note that this derivative is 1. 00:20:21
And if I get closer to the left, I would have to substitute in this other expression, 2 by 0, 0. 00:20:34
What we see is that the lateral derivative on the right and on the left does not coincide. 00:20:42
Therefore, there is no derivative in 0. 00:20:54
We have commented this on other occasions, and it is that if we have angular points, as occurs in this case, 00:21:01
the function is not going to be derivable. 00:21:10
So, let's recapitulate. 00:21:13
The function nx is equal to 3 was not continuous, 00:21:15
so it is not derivable. 00:21:22
That's why we have to study continuity before derivability. 00:21:25
And nx is equal to 0, there is no derivative in 0, 00:21:29
because if we study the lateral derivatives on the right 00:21:34
and 0 on the left give us different results. 00:21:38
That is, if I draw the slope of the tangent line to the x-squared curve, 00:21:42
I would get a slope of 0, 00:21:49
but if I draw it on this other side, on the left, 00:21:51
I would get slope 1. 00:21:55
That is, several slopes cannot exist. 00:21:58
If there is a derivative, it is unique. 00:22:01
So there is no derivative of 0. 00:22:03
f of x, therefore, is continuous in all r, minus in 3. 00:22:06
And it is derivable in all r, minus in 0 and in 3. 00:22:20
In 3 because the function is not continuous, in 0 because the lateral derivatives do not coincide, 00:22:32
and then the derivative does not exist. 00:22:38
Let's see one last example in which they give me a function in pieces, 00:22:40
in which some parameters a and b appear and they ask me how much a and b have to be worth 00:22:44
for the function to be continuous. We are going to analyze, as always, first what happens in the 00:22:50
open intervals. In this case we see that the function from minus infinity to minus 2 is defined 00:22:56
as a straight line, a polynomial function of grade 1, from minus 2 to 3 it is defined 00:23:07
with a constant function equal to 4, and from 3 to infinity the function is defined as another 00:23:15
slope-dependent line, ax-2. We first study the open intervals, from minus infinity to minus 2. 00:23:23
In this first interval, the function is continuous, and it is continuous because the function equal to 3x plus b is continuous in all r. 00:23:36
In the open interval from minus 2 to 3, the function also does not present any continuity problem, 00:23:54
is continuous also because y equal to 4 is continuous in all r. 00:24:01
And from 3 to infinity, f of x is continuous also 00:24:14
because y equal to x minus 2 is continuous in all r, 00:24:24
therefore it is going to be continuous in this open interval. 00:24:29
I want the function to be also optimal. 00:24:32
For this we are going to see what the function in minus 2 is worth. 00:24:36
And in minus 2 we have to substitute here. 00:24:42
Here it is defined for minus 2, because the image of minus 2 is worth 4. 00:24:46
The limit, when x tends to minus 2 on the right, 00:24:52
then I have to substitute, if I get closer to minus 2 on the right, 00:25:00
here in this function, which is going to be a constant, I have 4 left. 00:25:10
And the limit when x tends to minus 2 on the left, 00:25:18
in this case if I get closer to the left of minus 2, I have to take this other expression, 00:25:23
3x plus b. When substituting for minus 2, this would be minus 6 plus b. 00:25:32
I want the function to be continuous in x equal to minus 2, 00:25:41
therefore the three things have to exist and coincide. 00:25:47
In this case, 4 has to be equal to minus 6 plus b. 00:25:50
So from here we clear that b has to be worth 10. 00:25:56
Now let's see what happens in x equal to 3. 00:26:02
In x equal to 3, the function is defined for x equal to 3, it is worth 4. 00:26:05
So f of 3 is equal to 4 as well. 00:26:17
And we study the lateral limits by the right side of 3 and by the left side of 3. 00:26:25
If I approach the right side of 3, the expression I have to take for f would be ax-2. 00:26:30
And the result of substituting would be 3a-2. 00:26:44
The limit when x tends to 3 on the left, on the left of 3, the function f of x is defined as the constant function 4. 00:26:48
So this limit is 4. 00:27:04
And I want the function to be continuous 3. 00:27:06
So it has to be 3a-2, it has to be 4. 00:27:08
Resolving, I have to have a, it has to be 2. 00:27:14
In this case, I got a system of two uncoupled equations, 00:27:21
in the sense that from one of the equations I have cleared the b 00:27:26
and from another of the equations I have cleared the other parameter, the value of a. 00:27:30
Sometimes I get a system of equations that I have to solve. 00:27:35
Well, if a is 2 and b is 10, those are the values ​​that a and b have to take 00:27:38
for the function to be continuous throughout the row. 00:27:46
Idioma/s:
es
Idioma/s subtítulos:
en
Materias:
Matemáticas
Niveles educativos:
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  • Bachillerato
    • Primer Curso
    • Segundo Curso
Autor/es:
Francisca Florido Fernández
Subido por:
Francisca F.
Licencia:
Reconocimiento - No comercial - Compartir igual
Visualizaciones:
4
Fecha:
29 de julio de 2024 - 16:45
Visibilidad:
Clave
Centro:
IES ENRIQUE TIERNO GALVAN
Duración:
27′ 53″
Relación de aspecto:
1.78:1
Resolución:
1920x1080 píxeles
Tamaño:
64.51 MBytes

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