10 main principles of International law
Lecture 6: Function. Limits and continuity.
Plan:
1) Concept of function. Basic properties of functions.
2) Elementary functions. Classification of functions.
1) Concept of function. Basic properties of functions.
Definition 1. If to each element x of set X () is put in conformity the element y of set Y () speak, that on set X function is given. Where х is an independent variable (or argument), y - a dependent variable, and the letter f designates the law of conformity. Set X is a domain of function (or domain of existence), and set Y is a range of function. Methods of the task of functions:
1. Analytical method. Function is set by the formula . Example. .
2. Tabular method. Function is set by the table, containing values of argument х and corresponding values of function . Example. The table of logarithms.
3. Graphical method. Function is represented in the form of graph of function – of set on a plane, which abscisses are values of argument х, and ordinates - values of function corresponding them.
4. Verbal method. Function is described by a rule of its composition. Example. Function of Dirikhle: , if х - it is rational; , if х - it is irrational.
Basic properties of functions.
1. Parity and oddness. Function is even, if for any values x from a domain of function and odd, if . Otherwise function is a function of a general view.
Example. Function is even, as , function is odd, as . Function is function of a general view, as and , .
2. Monotony. Function is increasing (decreasing) on interval X if to greater value of argument from this interval corresponds greater (smaller) value of function.
Let and . Then function increases on interval X, if and decreases, if .
Increasing and decreasing functions is a monotonous functions.
Example. Function at decreases and at increases.
3. Limitation. Function is limited on interval X if there is such positive number , that for any . Otherwise function is unlimited.
Plan:
1) Concept of function. Basic properties of functions.
2) Elementary functions. Classification of functions.
1) Concept of function. Basic properties of functions.
Definition 1. If to each element x of set X () is put in conformity the element y of set Y () speak, that on set X function is given. Where х is an independent variable (or argument), y - a dependent variable, and the letter f designates the law of conformity. Set X is a domain of function (or domain of existence), and set Y is a range of function. Methods of the task of functions:
1. Analytical method. Function is set by the formula . Example. .
2. Tabular method. Function is set by the table, containing values of argument х and corresponding values of function . Example. The table of logarithms.
3. Graphical method. Function is represented in the form of graph of function – of set on a plane, which abscisses are values of argument х, and ordinates - values of function corresponding them.
4. Verbal method. Function is described by a rule of its composition. Example. Function of Dirikhle: , if х - it is rational; , if х - it is irrational.
Basic properties of functions.
1. Parity and oddness. Function is even, if for any values x from a domain of function and odd, if . Otherwise function is a function of a general view.
Example. Function is even, as , function is odd, as . Function is function of a general view, as and , .
2. Monotony. Function is increasing (decreasing) on interval X if to greater value of argument from this interval corresponds greater (smaller) value of function.
Let and . Then function increases on interval X, if and decreases, if .
Increasing and decreasing functions is a monotonous functions.
Example. Function at decreases and at increases.
3. Limitation. Function is limited on interval X if there is such positive number , that for any . Otherwise function is unlimited.