Dear participants, welcome to our lecture devoted to functions. These are the keywords of the lecture. We consider two type of functions. Functions of one variable and the functions of several variables. To study these functions, we need to discuss topology of the definition domain of functions, which is in our cases are set of real numbers and R. We have the space of dimension n, real vector space of dimension n. The usual topics related to these functions are continuity, derivatives, and integrals. First part is topology of R of the set of real numbers and continuity of functions of one variable. First, the general definition for mapping or transformation. Given two abstract sets, X and Y, we define mapping or transformation from X to Y as a rule that for any element, small x from the space X produces or generates an element y from the space Y. The element y is called image of x, and the mapping or transformation f, and x is a pre-image of the element y. The space X is sometimes called domain space or definition domain. The space Y is called image space of mapping or image domain, and we go to function when the space Y is a space of real numbers. In the case when big Y equal to R, we usually call the mapping or transformation function. This is a real functions. You see example of two functions of one real variables, two real functions. One is a quadratic functions, x squared plus 2, and the other one is well-known sigmoid function. You can also visit for functions of one variable, real functions so far variable as it is a very beautiful way to visualize such functions by graph of a function. You see two examples of graphs of these two functions. Now examples, so for functions of many variables, several variables. First example is the function is defined in the vector space of column vectors of dimension 2, and as a result of application of f to a vector is a Manhattan norm of this vector. This is a real function of two variables. Another example is a function of three variables defined in the space of column vector of dimension 3. This is a quadratic function of three variables. The last function in example is a function defined in the space of matrices. Dimension of this space is 3 multiplied by 3 equal 9. This function define a matrix determinant. This is explicit formula for the matrix determinant. Topology of R is from one side sample. From the other side, not so simple. You'll see or discuss a little while. First, we need to define a neighborhood of any element of R neighborhood of any real number. By definition, this is the Delta neighborhood of the point a is an interval, symmetric interval, with a center at the point a and these two extreme points a minus Delta and a plus Delta. Neighborhood of the points in the space of real numbers allows us to define a limit of a sequence of real numbers. For a given sequence, x^n of real numbers. Recall the point, a limit of the sequence. It is written, a equals limit x^n if for any positive Delta all elements of the sequence except a finite number of elements belong to the Delta neighborhood of the point a. It means that the elements of the sequence are concentrated around this point, the point a. The sequence which has a limit is called convergent sequence. We say sequence converge to sequence x^n converge to a. Two examples. First one is the example of convergent sequence and the limit of this sequence is equal to one. You can check, this may be good to exercise. You can check that for any neighborhood of the point 1, all elements of this sequence belongs to this neighborhood starting from some index. It is possible to write down explicitly what is this starting index, n of Delta. Another example is example of non-convergence sequence, which has no limit. This sequence is a minus 1 power n and the elements of the sequence are 1 minus 1,1 minus 1,1 minus 1. It is not maybe so easy but possible to check that there is no number a, which can be limit of this sequence, why? For example, a equal to 1 is not limit of the sequence because infinity many elements of the sequence y outside small neighborhood of this point. The same is for the point minus 1. If a is different from one and minus one, then you can find a neighborhood of this point where more elements of the sequence y. So there is no limit. The Convergent sequence give us possibility to define two type of sets. In this sets, we'll define the topology of the space of real numbers. One type of sets is open sets, another type is closed sets. Open set is a set that contains, it is each element of this set, some neighborhood of this element. The simplest example of open set in R is open interval. In fact, in R, the structure of open sets is rather simple. Any open set is a union of infinitely many in general, open disjoint intervals. Closed set in R is a set which contains all limits of all sequences from this set. The closed interval a, b is example of a closed set. But the structure of closed set in R is much more complicated than the structure of open sets. Famous example of really strange set is a well-known example of Cantor set. Anyway, some sets can be not open, not closed. For example, semi-closed interval, CD, is not closed because the point C is a limit of some points of this set, but it does not belong to this set and it is not open because each neighborhood of the point D has non-empty intersection. This is interval, but is not included in this interval. Problem that you can try to solve shows that finite set is always closed and never open. For a given set, show that this set is not closed and discuss how one can make it closed. Once we define limit open and the closed sets, we can define the continuous function of one variable. This is a very classical definition. Function is defined on the closed interval with the real values and it is continuous at a given point, x^0, if for any sequence x^n convergent to x^0, the values of functions f of x^n converge to the value of limit point f of x is 0. If it is true for any points in the interval, the function is called continuous function on the interval. Most of functions are continuous. Some easy examples of discontinuous function you'll see later on. But in general, they are not natural in some sense, they are looking like monster in the analysis of functions. Below you see examples of continuous functions. Everywhere, it is indicated the set or interval in which this function is continuous. Just try to check this property. Example of a discontinuous function, simplest example. If a sign function or signum function, which is equal to one for positive value of x^0 for x equals 0 and minus 1 for x negative. This function is not continuous at only one point, x^0 equal to 0. Why? Because, for example, one can take a sequence of positive numbers convergent to zero. The value of functions as a element of the sequence will be all equal to one and the limit of the sequence f of x_n will be equal to one. But this limit is not equal to the value of function at the point zero, which is zero. One of the most important property of continuous function on a closed interval is the well-known Weierstrass theorem. If function defined on a closed interval is continuous, then this function takes its maximum and minimum value. There exist two points, x_ min and x_ max, such that the value of function are all between f of x_ min and f of x_ max. This gives us existence theorem for extreme value of function and this theorem is a basic theorem in optimization. To show that a case of closed interval is important, we can see a counterexample. The function log of x is continuous on the open interval 0,2, but this function does not take minimum value on this interval because there is no minimum value of this function on this interval. Formally, minimum value is minus infinity.