1.8. Continuity of a function. Weierstrass theorem.

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From the course by National Research University Higher School of Economics

Mathematics for economists

6 ratings

National Research University Higher School of Economics

Mathematics for economists

6 ratings

This course is an important part of the undergraduate stage in education for future economists. It's also useful for graduate students who would like to gain knowledge and skills in an important part of math. It gives students skills for implementation of the mathematical knowledge and expertise to the problems of economics. Its prerequisites are both the knowledge of the single variable calculus and the foundations of linear algebra including operations on matrices and the general theory of systems of simultaneous equations. Some knowledge of vector spaces would be beneficial for a student. The course covers several variable calculus, both constrained and unconstrained optimization. The course is aimed at teaching students to master comparative statics problems, optimization problems using the acquired mathematical tools. Home assignments will be provided on a weekly basis. The objective of the course is to acquire the students’ knowledge in the field of mathematics and to make them ready to analyze simulated as well as real economic situations. Students learn how to use and apply mathematics by working with concrete examples and exercises. Moreover this course is aimed at showing what constitutes a solid proof. The ability to present proofs can be trained and improved and in that respect the course is helpful. It will be shown that math is not reduced just to “cookbook recipes”. On the contrary the deep knowledge of math concepts helps to understand real life situations.

From the lesson

The basics of the set theory. Functions in Rn

Week 1 of the Course is devoted to the main concepts of the set theory, operation on sets and functions in Rn. Of special attention will be level curves. Also in this week introduced definitions of sequences, bounded and compact sets, domain and limit of the function. Also from this week students will grasp the concept of continuous function.

1.1. Definitions and examples of sets14:35

1.2. Operations on sets9:31

1.3. Open balls in Rn9:10

1.4. Sequences in Rn. Closed sets.10:33

1.5. Bounded and compact sets10:48

1.6. Functions and level curves in Rn16:47

1.7. Domain and limit of a function. Continuous functions.13:51

1.8. Continuity of a function. Weierstrass theorem.13:04

1.9. Composite function.12:07

1.10. Continuity of a composite function3:30

Meet the Instructors

Kirill Bukin
Associate Professor, Candidate of sciences (phys.-math.)
Faculty of Economic Sciences, Department of Theoretical Economics

Now, I will slightly change the function from the previous example.

This time let f(x,y) be given by the formula,

when x times y is divided by the square root of x squared plus y squared.

Once again, since we're not allowed to divide by 0,

we exclude the origin.

The question is as before whether this particular function

is the limit when xy approach the origin.

Let me remind you that from Cartesian coordinates x and y,

we can change them

into polar coordinates by the formulas.

Let me draw a coordinate plane.

Suppose we have the point here,

whose coordinates are x and y.

Now, if we join this point with

the origin by a segment of a straight line or we can think this is a vector,

then this is an angle which we denote it by Phi letter, Greek letter Phi.

Now, r is the length of this vector.

So, r equals, from the right triangle,

the sum of the sides plus the length of

the hypotenuse and Phi is the angle drawn here.

Quite often, the change of the variables from Cartesian to polar

are quite useful and helpful in finding the limit.

Now when we replace x and y by the polar coordinates,

substituting into the arguments on the function f, we have f,

we get,

after simplifying what's left.

Now, what it means that x and y tend to origin,

x y tend to origin (0,0) and here we have f,

that means that r also tends to 0,

Now, here we have the function,

I can repeat here r tends to

0 and we have r times sine Phi cos Phi.

Now, recall from simple calculus that there is a Sandwich Theorem for the sequences,

this theorem claims that

given three sequences which

are in the double inequality,

sequences of real numbers xn, yn, and zn.

Let's suppose the very left sequence

tends to a and the very right tends to a,

now from this theorem it follows that yn,

which is in between that's why is called Sandwich Theorem,

also should tend to a.

We can use it here although we have no sequences here.

Still, we have a function which is a product of

the radius r which is becoming progressively smaller and smaller,

and we have a product of two trigonometric functions each of them is a bounded function.

We now Sine Phi as well as cosine Phi the values of

these functions never exceed positive one

and never are less than negative one and positive one,

so they lay in between.

That means that if I write

the absolute value

taking into account the boundedness of these functions I have simply are here,

and the absolute value is always non-negative and this tends to 0.

So, using the Sandwich Theorem or rather this analog,

we can say that the limit exists and it equals 0.

So, this function tends to 0.

Now, after this introduction about limits, let's define continuity

and we'll define this property of a function at a given point.

So, let's suppose we have

a function of n variables and D is domain,

and a particular point x0 belongs to domain.

Function f(x) is continuous

at this point if the limit of this function

exists as x approaches 0 and it equals f of (x0).

If function f(x) is continuous at

any point of D,

then we call it

continuous on D.

Continuous functions play a very important role in economic analysis

and this is the right time to recall the Weierstrass Theorem.

Remember I said that if a function is given and this function

is a function of one variable and is continuous on a closed interval,

so we're recalling the theorem from

single variable calculus,

hearing by Weierstrass.

Let f(x) be continuous on [a,b] closed interval,

then they exist numbers and such

that the function is bounded,

it's values lies in between lowercase m and capital M and

these values are attainable because f(x1) is

m and f(x2) is capital M.

The same theorem or rather similar theorem holds for

n-dimensional case but this time we have no segments in the n-dimensional space,

so we instead have compact sets.

So, if it's a statement you

replace the domain which is a segment by the compact set in rn,

then we can keep the rest.

So, this theorem claims that there will be two values, lowercase m,

capital M such that the value of the function f lies between these two values

and there exist two points from the domain x1x2.

This time we have to write instead of subscripts,

upscript such that the function at one point

takes the least value and in another point it takes the greatest value.

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